Skip to main content

Full text of "The Elements of Euclid; Sir Thomas Heath-2nd Ed. ebox Set"

See other formats


THE THIRTEEN BOOKS 
OF 

EUCLID'S ELEMENTS 



THE THIRTEEN BOOKS OF 
EUCLID'S ELEMENTS 

TRANSLATED FROM THE TEXT OF HEIBERG 

WITH INTRODUCTION AND COMMENTARY 



Sir THOMAS L. HEATH, 

K.C.B., K.C.V.O., F.R.S., 

SC.D. CAMS., HON. D.SC. OXFORD 
HONORARY FELLOW (SOMETIME FELLOW) OF TRINITY COLLEGE CAMBRIDGE 



SECOND EDITION 

REVISED WITH ADDITIONS 

VOLUME n 
BOOKS III— IX 



DOVER PUBLICATIONS, INC. 

NEW YORK 



J ^A'JV^' < ]^[ >Mi-j i ii 



I ' r, I I 



'III I , 



Thl( new edlHon, Uttt publlihed in 1956, U an 
un&b ridged and unaltered lepublicatton of tJie 
lecond edition. It It publlthed through ipeclal 
airoogement with Cambridge Unlvcnity Preti. 



Librajy of Congress Catalog Card Numbers ^6-4)36 



Manufactured in the United States of Amerioi 

Dover Publicaciona, Inc. 
180 Varick Street 
■^ New York 14, N. Y. 



CONTENTS OF VOLUME 11. 



Book 


III. 


Definitions 


I 






Propositions ..... 


6 


Book 


IV. 


Definitions 


. . 78 






Propositions 


80 


Book 


V. 


Introductory note 


1 1 z 






Definitions 


U3 


* 




Propositions 


• ■ 138 


Book 


VI. 


Introductory note 


. . 187 






Definitions 


188 






Propositions 


i9r 


Book 


VII. 


Definitions 


«77 






Propositions 


296 


Book VIII. 


, 


345 


Book IX. 




384 


Greek Index to Vol. II 


417 


English Index to Vol. II 


431 



BOOK III. 



DEFINITIONS. 

[. £qual circles are those the diameters of which are 
equal, or the radii of which are equal. 

2. A straight iine ts said to touch a circle which, 
meeting the circle and being produced, does not cut the 
circle, 

3. Circles are said to touch one another which, 
meeting one another, do not cut one another. 

4. In a circle straight lines are said to be equally 
distant from the centre when the perpendiculars drawn 
to them from the centre are equal. 

5. And that straight line is said to be at a greater 
distance on which the greater perpendicular falls. 

6. A segment of a circle is the figure contained by a 
straight line and a circumference of a circle. 

7. An angle of a segment is that contained by a 
straight line and a circumference of a circle. 

8. An angle in a segment is the angle which, when 
a point is taken on the circumference of the segment and 
straight lines are joined from it to the extremities of the 
straight line which is the base of the segment, is contained 
by the straight lines so joined. 

mi . . .. 

9. And, when the straight lines containing the angle cut 
off a circumference, the angle is said to stand upon that 

circumference, 



» BOOK in [hi. deff. 

10. A sector of a circle is the figure which, when an 
angle is constructed at the centre of the circle, is contained by 
the straight lines containing the angle and the circumference 
cut off by them. 

11. Similar segments of circles are those which 
admit equal angles, or in which the angles are equal to one 
another. 

Definition i. 

Iiroi kvkXkh turivi wy at Stdfitrpoi urai tltrtVj ^ mv al fK rur HtrTptnv ivai tUrir, 

Many editors have held that this should not have been included among 
deAnitions. Some, e.g. Tartaglia, would call it apos/u/aU; others, e.g. Borelli 
and Playfair, would c^l it an axiom ; others again, as Billingsley and Clavius, 
while admitting it as a definitien, add explanations based on the mode of 
constructing a circle ; Simson and Pfleiderer hold that it is a tfuoretn, I 
think however that Euclid would have maintained that it is a definition in 
the proper sense of the term ; and certainly it satisfies Aristotle's requirement 
that a "definitional statement" (opurTdtM AoyM) should not only state the 
fait (to iri) but should indicate the cause as well {De aitima ii. i, 413 a 
13). The equality ot circles with equal radii can of course be proved by 
superposition, but, as we have seen, Euclid avoided this method wherever he 
could, and there is nothing technically wrong in saying " By equal circles 1 
mean circles with equal radii." No flaw is thereby introduced into the system 
of the Elements ; for the definition could only be objected to if it could be 
proved that the equality predicated of the two circles in the definition was 
not the same thing as the equality predicated of other equal figures in the 
Elements on the l^is of the Congruence- Axiom, and, nt^less to say, this 
cannot be proved because it is not true. The existence of equal circles (in 
the sense of the definition) follows from the existence of equal straight tines 
and I. Post. 3, 

The Greeks had no distinct word for radius, which is with them, as here, 
the {straight line drawn) from the centre 7 Jk r™ mVrpou ((Wiln) ; and so 
definitely was the expression appropriated to the radius that in tov Kit^pou 
was used without the article as a predicate, just as if it were one word. Thus, 
eg., in III. I JK KtvTpov yap means " for they are radii " : cf, Archimedes, On 
the Sphere and Cylinder i\. z, ij BE Ik rm xiirpoa iarl Tm,,,KiK^jm, BM is it 
radius of the circle. 

Definition 2. 

Euclid's phraseology here shows the regular distinction between ainvr$vii 
and its compound li^maOai, the former meaning "to tnett" and the latter 
"to touch." The distinction was generally observed, by Greek geometers 
from Euclid onwards. There are however exceptions so far as hrrfaBm is 
concerned; thus it means "to touch" in Eucl. iv. Def 5 and sometimes in 
Archimedes On the other band, ^c/iairTccrdat is used by Aristotle in certain 



III. DEFF. i— 4] DEFINITIONS 3 

cases where the orthodox geometrical term would be airrifrtfot. Thus in 
Meleerohgica m. 5 (376 b 9) he says a certain circle will pass through all the 
angles (ajTotrulK i^nu^tTM t^v yioi'tiui'), atid (376 a 6) M will lie on a given 
(circular) circumference {iihofian)^ trtpti^tptiiK c^'i^crat to M). We shall find 
awrviBai used in these senses in Book iv. Deff. 2, 6 and Deff. 1, 3 respectively. 
The latter of the two expressions (quoted from Aristotle means that thi locus 
of M is a given drde, just as in Pappus o^trai to armiiov Oia-n StSo/tiyrft 
ti$nat means that th^ locus ^the point is a straight line given in position. 

Definition 3. 

Todhunter remarks that different opinions have been held as to what is, 
or should be, included in this definition, one opinion being that it only means 
that the circles do not cut in the neighbourhood of the point of contact, 
and that it must be shown that they do not cut elsewhere, while another 
opinion is that the definition means that the circles do not cut at all 
Todhunter thinks the latter opinion correct. I do not think this is proved ; 
and I prefer to read the definition as meaning simply that the circles meet 
at a point but do not cut at that point. I think this interpretation 
preferable for the reason that, although Euclid does practically assume in 
III. ti — :3, without stating, the theorem that circles touching at one point 
do not intersect anywhere else, he has given us, before reaching that 
point in the Book, means for proving for ourselves the truth of that 
statement. In particular, he has given us the propositions in. 7, 8 which, 
taken as a whole, give us more information as to the general nature of a 
circle than any other propositions that have preceded, and which can be used, 
as will be seen in the sequel, to solve any doubts arising out of Euclid's 
unproved assumptions. Now, as a matter of fact, the propositions are not used 
in any of the genuine proofs of the theorems in Book in. ; in. 8 is required 
for the second proof of ni. 9 which Simson selected in preference to the first 
proof, but the first proof only is regarded by Heibecg as genuine. Hence it 
would not be easy to account for the appearance of in. 7, 8 at all unless as 
affording means of answering possible objections {cf. Proclus' explanation of 
Euclid's reason for inserting the second part of i. 5). 

External and internal contact are not distinguished in Euclid until 111. 
II, 12, though the^^w of in. 6 (not the enunciation in the original text) 
represents the case of internal contact only. But the definition of touching 
circles here given must be taken to imply so much about internal and external 
contact respectively as that (a) a circle touching another internally must, 
immediately before " meeting " it, have passed through points within the 
circle that it touches, and {b) a circle touching another externally must, 
immediately before meeting it, have passed through points outside the circle 
which it touches. These facts must indeed be admitted if internal and 
external are to have any meaning at all in this connexion, and they constitute 
a minimum admission necessary to the proof of in. 6. 

Definition 4. 

'Ev kukA^ urof Ltck-^w airo roC Kci'Tpou cf^tuii AryotTui, orat' at a.vh TOv 
Ktyrpov iv aitras Kti0tTQi oiyOfm^t i(7u4 ciKTti^. 



BOOK III [in. DEFF. 5—9 

Definition s- 



Definition 6. 

T/t^/ia KuicXov hm to wtpit^ofifyoy (rj^^/ia iro n (v^ttat Hat kvkXov 

- .. Definition 7. 

TfHj/iOTOS 8c yMwla tirrlv jj ircpit)(onfirr) v'lni t* iJS«w »«ii KUtkov wtfyi^ptlat, 
ThU definition is only interesting historically. The an^& of a segment, 
being the " angle " formed by a straight line and a " circumference," is of the 
kind described by Proclus as " mixed." A particular " angle " of this sort is 
the "angle of a semicircle," which we meet with again in ui. i6, along with 
the so-called "horn -like angle" (jMparmtSijs), the supposed "angle" between 
a tangent to a circle and the circle itself. The " angle of a semicircle " occurs 
once in Pappus (vii. p. 670, 19}, but tt there means scarcely more than the 
corner of a semicircle regarded as a point to which a straight line is directed. 
Heron does not give the definition of the att£k of a segment, and we may 
conclude that the mention of it and of the angle of a umieircle in Euclid is a 
survival from earlier text-books rather than an indication that Euclid considered 
either to be of importance in elementary geometry (cf. the note on iii- i6 
below). 

We have however, in the note on i. s above (Vol. 1, ppi 252—3), seen evi- 
detice that the a»^& ^aj<^Mf had played some part in geometrical proofs up 
to Euclid's time. It would appear from the passage of Aristotle there quoted 
{Anal, prior, i, 24, 41 b 13 sqq.) that the theorem of 1. 5 was, in the text-books 
immediately preceding Euclid, proved by means of the equality of the two 
" angles of" any one segment. This latter property must therefore have been 
regarded as more elementary (for whatever reason) than the theorem of i. 5 ; 
indeed the definition as given by Euclid practically implies the same thing, 
since it speaks of only one " angle of a segment," namely "/At angle contained 
by a straight line and a circumference of a circle," Euclid abandoned the 
actual use of the "angle" in question, but no douht thought it unnecessary 
to bieak with tradition so far as to strike the definition out also. 



Definition 8. 

tnffutoy nal air a^oi> ittI Tft ir^ara tt^^ tvOtiti^, 7f itrrt ^ovif TOtJ TfiT/jfj^aro^f 



Definition 9. 

'Orav Bt at vepiij(owTat r^v yuiviav eo&tiai dirakafi^tii'tiMri xim irtpn^tpiiav, 



III. DErr. it^ ii] NOTES ON DEFINITIONS 5—11 J 

Definition 10. 

To/uvt Si icvK\m/ itrriv, Smv irpof ry ithirp^ rov NtLncXou murra^ yaria, 
TO n*pttx^firvotf <r)mfia into re Tmv TTfr ytaviav vtptt^ov^r^v (u^ttwv tcai n^ 
djroXa/A^tLvoftivjjt vtt* airrSv vtpu^*p<itK. 

A scholiast says that it was the shoemaket's knife, trtarrvrti^uA^ roiitit, 
which su^ested the name ra^t for a. sector of a circle. The derivation of 
the name from a resemblance of shape is parallel to the u^ of ap^irXos (also 
a sMetmakfr't knife) to denote the well known figure of the Book of Lemmas 
partly attributed to Archimedes. 

A wider definition of a sector than that given by Euclid b found in a 
Greek scholiast (Heiberg's Euclid, Vol. v. p. 260) and in an-Nairizi (ed. Curtze, 
p. hi). "There are two varieties of sectors ; the one kind have the angular 
vertices at the centres, the other at the circumferences. Those others which 
have their vertices neither at the circumferences nor at the centres, but at 
some other points, are for that reason not called sectors but sector-like 
figures (td/m«(8ij v-x^partk)," The exact agreement between the scholiast and 
an-NairizI suggests that Heron was the authority for this explanation. 

The Mctor-Hkt figure bounded by an arc of a circle and two lines drawn 
from its extremities to meet at any point actually appears in Euclid's book On 
divisions {trtpl Sinifiwtuiv) discovered in an Arabic MS. and edited by 
Woepcke (cf. Vol. 1. pp. 8—10 above). This treatise, alluded to by Proclus, 
had for its object the division of figures such as triangles, trapezia, 
([uadrilaterals and circles, by means of straight lines, into parts equal or 
in given ratios. One proposition e.g. is, Ta divide a triangle into two equal 
parts by a straight lint passing through a given point on one side. The 
proposition (28) in which the quasi-udor occurs is, To divide suth a figure by a 
straight line into two equal parts. The solution in this case is given by Cantor 
(Gesck d. Math, u, pp. aS;— 8). 

If ABCD be the given figure, E the middle point 
of BD and EC at right angles to BD, 
the broken line AEC clearly divides the figure into 
two equal parts. 

Join AC, and draw EF parallel to it meeting 
AB\n F. 

Join CF, when it is seen that CF divides the 
figure into two equal parts. 

Definition u, ' - 

De Morgan remarks that the use of the word similar in "similar 
segments " is an anticipation, and that similarity of form is meant. He adds 
that the definition is a theorem, or would be if " similar " had taken its final 
meaning. 




BOOK III. PROPOSITIONS. 




Proposition i. 

To find the centre of a given circle. 

Let ABC be the given circle ; 
thus it is required to find the centre of the circle ABC. 

Let a straight line AB be drawn 
s through it at random, and let it be bisected 
at the point D ; 

from D let DC be drawn at right angles 
to AB and let it be drawn through to E ; 
let CE be bisected at F\ 
"o I say that F is the centre of the circle 
ABC. 

For suppose it is not, but, if possible, 
let G be the centre, 

and let GA, GD, GB be joined. ,., , .. , 

IS Then, since AD is equal to DB, 
and DG is common, 

the two sides AD, DG are equal to the two sides 
BD, DG respectively ; 

and the base GA is equal to the base GB, for they are 
20 radii ; 

therefore the angle ADG is equal to the angle GDB. [i. 8] 

But, when a straight line set up on a straight line makes 

the adjacent angles equal to one another, each of the equal 

angles is right ; [i. Def, 10] 

35 therefore the angle GDB is right. 



111. i] PROPOSITION I 7 

But the angle FDB is also right ; 
therefore the angle FDB is equal to the angle GDB, the 
greater to the less : which is impossible. 

Therefore G is not the centre of the circle ABC. 
30 Similarly we can prove that neither is any other point 
except F. 

Therefore the point F is the centre of the circle ABC. 

, PoRiSM. From this it is manifest that, if in a circle a 

straight line cut a straight line into two equal parts and at 

35 right angles, the centre of the circle is on the cutting straight 
line. 

Q. E. F. 

a. For auppose It is not. Tliis is expressetl in the Greek by the two wocds H^i yif, 
but Biicb an EllLpticaJ phrase is impossible in English. 

17. the two sides AD, DG are etjual to the two aides BD, DO respectively. 
As before observed, Euclid 13 not always oarerul to put the equals in correspond Itig order. 
The (est here has " CZ>, D3." 

'Fodhunter observes that, when, in the construction, DC is said to be 
productd to J?, it is assumed that D is within the circle, a fact which Euclid 
first demonstrates in in. 2. This is no doubt true, although the word iai^^w, 
" let it be drawn through^' is used instead of iK^ijiKi^trSm, " let it be^rcJuad." 
And, although it is not necessary to assume that I> is within the circle, it is 
necessary for the success of the construction that the straiglit line drawn 
through jD at right angles to AB shall meet the circle ir_ two points (and no 
more): an assumption which we are not entitled to make on the basis of what 
has gone before only. 

Hence there is much to be said for the alternative procedure recommended 
by De Morgan as preferable to that of Euclid. De Morgan would first prove 
the fundamental theorem that "the line which bisects a chord perpendicularly 
must contain the centre," and then make ni. i, iii. 25 and iv. 5 immediate 
corollaries of it. The fundamental theorem is a direct consequence of the 
theorem that, if P is any point equidistant from A 
and .5, then P lies on the straight line bisecting AJ3 
perpendicularly. We then take any two chords AB, 
j4Cof the given circle and draw £>0, EO bisecting 
them perpendicularly. Unless BA^ AC are in one 
straight line, the straight lines DO, EO must meet 
in some f)oint O (see note on iv. 5 for possible 
methods of proving this). And, since both DO, 
EO must contain the centre, must be the centre. 

This method, which seems now to be generally 
preferred to Euclid's, has the advantage of showing 

that, in order to find the centre of a circle, it is sufficient to know three points 
on the circumference. If therefore two circles have three points in common, 
they must have the same centre and radius, so that two circles cannot have 
three points in common without coinciding entirely. Also, as indicated by 
De Morgan, the same construction enables us (i) to draw the complete circle 
of which a segment or arc only is given {ill. 25), and (2) to circumscribe a 
circle to any triangle (iv. 5). 




8 BOOK. Ill [ill. I. 1 

But, if the Greeks had used this construction for finding the centre of a 
circle, they would have considered it necessary to add a proof that no other 
point than that obtained by the construction can be the centre, as is clear 
both from the similar rtduetio ad abturdum in iii i and also from the fact 
that Euclid thinks it necessary to prove as a separate theorem (ui. 9) that, if 
a point within a circle be such that three straight lines (at least) drawn from it 
to the circumference are equal, that point must be the centre. In fact, 
honrever, the proof amounts to no more than the remark that the two 
perpendicular bisectors can have no more than one point common. 

And even in De Morgan's method there is a yet unproved assumption. 
In order that DO, EO may meet, it is necessary that AB, AC should not be 
in one straight line or, in other words, that BC should not pass through A. 
This results from iii. 2, which therefore, stKctly speaking, should precede. 

To return to Euclid's own proposition HI. i, it will be observed that the 
demonstration only shows that the centre of the circle cannot lie on either 
side of CD, so that it must lie on CD or CD produced. It is however taken 
for granted rather than pioved that the centre must be the middle point of 
CE. The proof of this by rtduetio ad absurdum is however so obvious as to 
be scarcely worth giving. The same consideration which would prove it may 
be used to show that a circle cannot have more than one ctntre, a proposition 
which, if thought necessary, may be added to iii. i as a corollary. 

Simson o^rved that the proof of [ii. i could not but be by reductio ad 
aksurdum. At the beginning of Book in. we have nothing more to base the 
proof upon than the dejinitton of a circle, and this cannot be made use of 
unless we assume some point to be the centre. We cannot however assume 
that the point found by the construction is the centre, because that is the 
thing to be proved. Nothing is therefore left to us but to assume that some 
other point is the centre and then to prove that, whatever other point is 
taken, an absurdity results; whence we can infer that the point found is 
the centre. 

The Porism to in. i is inserted, as usual, parenthetically before the words 
Svtp I5» T«7<7at, which of course refer to the problem itself. 

Proposition 2. 

If on the circumference of a circle two points be taken at 
random, the straight line joining the points will fall within 
the circle. 

Let ABC be a circle, and let two points A, B \x. taken 
at random on its circumference ; . 

I say that the straight line joined from 
.^ to Z? will fall within the circle. 

For stippose it does not, but, if 
possible, let it fall outside, as AEB ; 
let the centre of the circle ABC be 
taken [in. 4 and let it be Z? ; let DA, 
DB be joined, and let DFE be drawn 
through. 




III. i] PROPOSITIONS I, J 9 

• Then, since DA is equal to DB, 

the angle DAE is also equal to the angle DBE. [i. 5] 
And, since one side AEB of the triangle DAE is produced, 
the angle DEB is greater than the angle DAE. [i. 16] 
But the angle DAE is equal to the angle DBE ; 
therefore the angle DEB is greater than the angle DBE. 
And the greater angle is subtended by the greater side ; [i. 19] 
therefore DB is greater than DE. 
But DB is equal to DF\ ' ~ - 

therefore DF is greater than DE, 

the less than the greater : which is impossible. 
Therefore the straight Une joined from A to ^ will not 
fall outside the circle. 

Similarly we can prove that neither will it fall on the 
circumference itself; 

therefore it will fall within. ,< , 
Therefore etc. 

• • Q. E. D. 

The nduitio ad absurdum form of proof is not really necessary in this case, 
and it has the additional disadvantage that it requires the destruction of two 
hypotheses, namely (hat the chord is (i) outside, (i) on 
ihe circle. To prove the proposition directly, we have 
only to show that, if ^ be any point on the straight line 
AB between A and B, DE is less than the radius of the 
circle. This may be done by the method shown above, 
under i. 24, for proving what is assumed in that 
proposition, namely that, in the hgurc of the proposition, 
/"falJs beiow EG if DE is not greater than DF. The 
assumption amounts to the following proposition, which 
De Morgan would make to precede 1. ^4 ; " Every 
straight line drawn from the vertex of a triangle to the base is less than 
the greater of the two sides, or than either if they be equal." The case 
here Is that in which the two sides are equal ; and, since the angle DAB is 
equal to the angle DBA, while the exterior angle DEA is greater than the 
interior and opposite angle DBA, it follows that the angle DEA is greater 
than the angle DAE, whence DE must be less than DA or DB. 

Camerer points out that we may add to this proposition the further 
statement that all points on AB produced in either direction are outside the 
circle. This follows from the proposition (also proved by means of the 
theorems that the exterior angle of a triangle ts greater than either of the 
interior and opposite angles and that the greater angle is subtended by 
the greater side) which De Morgan proposes to introduce after i. 3 1, namely, 

" The perpendicular is the shortest straight line that can be drawn from a 




10 



BOOK in 



[ill. 2, 3 



given p^oint to a given straight line, and of others that which is nearer to the 
perpendicular is less than the more lemote, and the converse ; also not more 
than two equaJ straight lines can be drawn from the point to the line, one on 
each side of the perpendicular." 

The fact that not more than two equal straight lines can be drawn from a 
given point to a given straight line not passing through it is proved by Proclus 
on ], 1 6 (see the note to that proposition) and can alternatively be proved by 
means of i. 7, as shown above in the note on I. 1 2. It follows that 

A straight line cannot cut a circle in Men than two points 
a proposition which De Morgan would introduce here after in. a. The proof 
given does not apply to a straight line passing through the centre j but that 
!iiich a line only cuts the circle in two points is self evident 



Proposition 3. 

If in a circle a straight line through the centre bisect a 
straight line not through the centre, it also cuts it at right 
angles : and if it cut it at right angles, it also bisects it. 

Let ABC be a circle, and in it let a straight line CD 
J throi]gh the centre bisect a straight line 
AB not through the centre at the point 
F; 

I say that it also cuts it at right angles. 
For let the centre of the circle ABC 
10 be taken, and let it be .£"; let EA, EB 
be joined. 

Then, since AF is equal to FB, 
and FE is common, 

two sides are equal to two sides ; 
IS and the base EA is equal to the base EB ; 

therefore the angle AFE is equal to the angle BFE. [1.8] 

Biit, when a straight line set up on a straight line makes 

the adjacent angles equal to one another, each of the equal 

angles is right ; [i- Def. 10] 

20 therefore each of the angles AFE, BFE is right. 

Therefore CD, which is through the centre, and bisects 
AB which is not through the centre, also cuts it at right 
angles. 

Again, let CD cut AB at right angles ; 
«5 I say that it also bisects it. that is, that AFis equal to FB. 




in. 3. 4] PROPOSITIONS 2—4 11 

For, with the same construction, 

since £A is equal to EB, 
the angle EAF is also equal to the angle EBF. [1. 5] 

But the right angle AFE is equal to the right angle BEE, 
JO therefore EAF, EBF are two triangles having two angles 
equal to two angles and one side equal to one side, namely 
EF, which is common to them, and subtends one of the equal 
angles ; 

therefore they will also have the remaining sides equal to 
35 the remaining sides ; [i- 26] 

therefore AF is equal to FB. 
Therefore etc. 

Q. E. D. 

46. with the eame construction^ Ttav ntW-up jrnracrjrFLacr^^rrup . 

This proposition asserts the two partial converses (cf, note on i. 6) of the 
Porism to iti. i. De Morgan would place it next to in. i. 

pROPOSITiON 4. 

If in a circle two straight lines cut one another which are 
not through the centre, they do not bisect one another. 

Let A BCD be a circle, and In it let the two straight lines 
AC, BD, which are not through the 
centre, cut one another at E \ 
I say that they do not bisect one 
another. 

For, if possible, let them bisect one 
another, so that AE is equal to EC, 
and BE to ED ; 

let the centre of the circle ABCD be 
taken [in, 1], and let it h^ F\ let FE be 
joined. 

Then, since a straight line FE through the centre bisects 
a straight line AC not through the centre, 

it also cuts it at right angles ; [ni, 3] 

therefore the angle FEA is right. 
Again, since a straight line FE bisects a straight line BD, 
it also cuts it at right angles ; fm. 3] 

therefore the angle FEB is right. 




xa 



BOOK III 



[ill. 4, 5 



But the angle FEA was also proved right ; 
therefore the angle FEA is equal to the angle FEB, 
the less to the greater : which is impossible. 

Therefore AC, BD do not bisect one another. 
Therefore etc. 

y. K. D. 




Pkovosition 5. 

1/ two circles cut one another, they will not have ike same 
centre. 

For let the circles ABC, CDG cut one another at the 
points B, C\ 

I say that they will not have the same 
centre. 

For, if possible, let it be E \ let EC 
be joined, and let EFG be drawn 
through at random. 

Then, since the point E is the 
centre of the circle ABC, 

EC is equal to EF. [i. Def. 15] 

Again, since the point E is the centre of the circle CDG, 
EC is equal to EG. 

But EC was proved equal to iS'/'also ; 

therefore EF is also equal to EG, the less to the 
greater : which is impossible. 

Therefore the point E is not the centre of the circles 
ABC, CDG. 

Therefore etc. ' . ' . 

Q. E. D. 

The propositions nt. 5, 6 could be combined in one. It makes no 
difference whether the circles cut, or meet without cutting, so long as they do 
not coincide altogether; in either case they cannot have the same centre. 
The two cases are covered by the enunciatiorv ; If the circumferexces of two 
ctTctti meet at a point th^ {annot have the same centre. On the other hand, If 
two circles have the same autre and one point in their circumferences common, 
they mitst coincide altogether. 



III. 6] 



PROPOSITIONS 4—6 



n 



Proposition 6. 

1/ two cinles touch one another, they will not have the 
same centre. 

For let the two circles ABC, CDE touch one another 
at the point C\ 

I say that they will not have the 
same centre. 

For, if possible, let it be F; let 
FC be joined, and let FEB be drawn 
through at random. 

Then, since the point F is the 
centre of the circle ABC, 

FC is equal to FB. 

Again, since the point F is the 
centre of the circle CDE, 

FC is equal to FE. „ 

But FC was proved equal to FB ; 

therefore FE is also equal to FB, the less to the greater: 
which is impossible. 

Therefore F is not the centre of the circles ABC, CDE. 
Therefore etc, 

Q. E, D. 




The English editions enunciate this propusltion of circles touching 
inttmaliy, but the word (<vt<k) is a mere interpolation, which was no doutrt 
made because Euclid's figure showed only the case of internal contact. The 
fact is that, in his usual manner, he chose for demonstration the more difficult 
case, and left the other case (that of external contact) to the intelligence of 
the reader. It is indeed sufficiently self-evident that circles touching extemally 
cannot have the same centre ; but Euclid's proof can really be used for thia 
case too. 

Camerer remarks that the proof of iti. 6 seems to assume tacitly that the 
points E and B cannot coincide, or that circles which touch internally at C 
cannot meet in any other point, whereas this fact is not proved by Euclid till 
jii. 13. But no such general assumption is necessary here; it is only 
necessary that one Une drawn from the assumed common centre should meet 
the circles in different points; and the very notion of internal contact requires 
that, before one circle metis the other on its inner side, it must have passed 
through points within the latter circle. 



1^4 ^ BOOK Ul (|n^«f 

-•'"•■ Proposition 7. 

// OK the diameter of a circle a point be taken which is not 
the centre of the circle, and front the point straight lines fall 
upon the circle, (hat will be greatest on which the centre is, tlie 
remainder of the same diameter will be least, and of ike rest 
5 the nearer to the straight line through the centre is always 
greater than the more remote, and only two equal straight 
lines will fall from the point on the circle, one on each side 
of ike least straight line. 

Let ABCD be a circle, and let AD be a diameter of it ; 
10 on v4Z) let a point F be taken which is not the centre of the 
circle, let E be the centre of the circle, 

and from F let straight lines FB, FC, FG fall upon the circle 
ABCD; 

I say that FA is greatest, FD is least, and of the rest FB is 
IS greater than FC, and FC than FG. 
For let BE, CE, GE be joined. 
Then, since in any triangle two 
sides are greater than the remaining 
one, [i- 20] 

ao EB, EF are greater than BF. 

But AE is equal to BE ; 
therefore AF\% greater than BF. 
Again, since BE is equal to CE^ 
and FE is common, ' ' " ' 

25 the two sides BE, EF are equal to the two sides CE, EF. 
But the angle BEFh also greater than the angle CEF; 
therefore the base BF is greater than the base CF. [i. 24] 

For the same reason 

CF is also greater than FG, , , ,. 

30 Again, since GF, FE are greater than EG, 
and EG is equal to ED, w- - • . 

GF, FE are greater than ED. 
Let EE be subtracted from each ; 

therefore the remainder GF is greater than the remainder 
i%FD. 

Therefore FA is greatest, FD is least, and FB is greater 
than FC, and FC than FG. 




I III. 7] PROPOSITION 7 *S 

1 say also that from the point F only two equal straight 
lines will fall on the circle A BCD, one on each side of the 
40 least FD, 

For on the straight line EF, and at the point E on it, let 
the angle ^^//^ be constructed equal to the angle GEF\\- aal. 
and let FH be joined. 

Then, since GE is equal to EH, 
4i and EF is common, 

the two sides GE, EF are equal to the two sides HE, EF; 
and the angle GEF is equal to the angle HEF ; 

therefore the base FG is equal to the base FH. [i- 4] 
I say again that another straight line equal to FG will no; 
so fall on the circle from the point F. 
For, if possible, let FK so fall. 
Then, since FK is equal to FG, and FH to FG, 
' tJ,ii. • FK is also equal to FH, -^n-i. 

the nearer to the straight tine through the centre being 
ss thus equal to the more remote : which is impossible. 

Therefore another straight line equal to GF m\\ not fall 
from the point F upon the circle ; 

therefore only one straight line will so fall. 
Therefore etc. 
'' Q. E. D, 

4, of the same diameter. I have iriseite<i Ihese words Tot clearness* sake. The text 
Kas simply ^Xax^ri^ A^ ij X«ir^, " and the remaining (straight line) least." 

7, 39. one on CAcb side. The word *' one * is not in the Greek, but is necessary to 
g:iv< the force of t^' hdrtpa r^ iKaxirnp, literally " on both sides," or " on each nrtlie two 
sides, of the leajit." 

De Morgan points out that there is an unproved assumption in this 
tietnonstration. We draw straight lintjs from F, as FB, FC, such that the 
angle DFB is greater than the angle DFC and then assume, with respect to 
the straight lines drawn from the centre E to B, C, that 
the angle DEB is greater than the angle DEC. This 
Is most easily pitived, I think, by means of the converse 
of part of the theorem about the lengths of different 
straight lines drawn to a given straight tine from an 
external point which was mentioned above in the note 
on III. J. This converse would be to the effect that, 1/ 
two unequal straight lines be drawn from a point to a 
gitxn straight line whieh are not perpendicular to the 
straight line, tht greater of the hm is tht further from the perfettdicular from the 
point to the given straight line. This can either be proved from its converse by 
rtductio ad absurdum, or established directly by means of i. 47. Thus, in the 
accompanying figure, FB must cut .£C in some point M, since the angle BFE 
is less than the angle CFE. 

Therefore EM is less than EC, and therefore than EB. tuff 




t« BOOK III [in. 7 

Hence the point B in which FB meets the cticie is further from the foot 
of the petpendicuUr from E on FB than i£'\% ; 

therefore the angle BEF\% greater than the angle CEF, 

Another way of enunciating the first part of the proposition is that of 
Mr H. M. Taylor, viz. " Of all straight lines drawn to a circle from an internal 
point not the centre, the one which passes through the centre is the greatest, 
and the one which when produced passes through the centre is the least; and 
of any two others the one which mbUnds the greater angle at the centre is the 
greater." The substitution of the angle subtended at the antre as the criterion 
no doubt has the effect of avoiding the necessity of dealing with the unproved 
assumption in Euclid's proof referred to above, and the similar substitution in 
the enunciation of the first part of i[i. 8 has the effect of avoidmg the necessity 
tor dealing with like unproved assumptions in Euclid's proof, as well as the 
complication caused by the distinction in Euclid's enunciation between lines 
falling from an external point on the convex eircumfercnce and on the ccneave 
dreumfe'rence of a circle respectively, terms which are not defined but taken as 
understood. 

Mr Nixon {Euclid Eevised) similarly substitutes as the criterion the angle 
subtended at the centre, but gives as his reason that the words " nearer " and 
" more remote " in Euclid's enunciation are scarcely clear enough without 
some definition of the sense in which they are used, Smith and Bryant make 
the substitution in iii. 8, but follow Euclid in lii. 7. 

On the whole, 1 think that Euclid's plan of taking straight lines drawn from 
the point which is not the centre direct to the circumference and making 
greater or less angles at that point with the straight line containing it and the 
centre b the more instructive and useful of the two, since it is such lines 
drawn in any manner to the circte from the point which are immediately useful 
in the proofs of later propositions or in resolving difficulties connected with 
those proofs. 

Heton again (an-Nairizi, ed, Curtze, pp. 114^5) ^^^ * "o'^ o" ''*'* 
proposition which is curious. He first of alt says that Euclid proves that lines 
nearer the centre are greater than those more remote fi'om it. This is a 
different view of the question from that taken in Euclid's proposition as we 
have it, in which the lines are not nearer to and more remote from the centre 
but from the line through the centre. Euclid takes lines inclined to the latter 
line at a greater or less angle ; Heron introduces distance from the centre in 
the sense of Deff. 4, 5, i.e. in the sense of the length of the perpendicular drawn 
to the line from the centre, which Euclid does not use till iii, t4, 15. Heron 
then obsen'es that in Euclid's proposition the lines compared are all drawn on 
one side of the line through the centre, and sets himself to prove the same 
truth of lines on opposite sides which are more or less distant ^iww the centre. 
The new point of view necessitates a quite different line of proof, anticipating 
the methods of later propositions. 

The first case taken by Heron is that of two straight lines such that the 
perpendiculars from the centre on them fall on the lines themselves and not 
in either case on the line produced. 

Let A be the given point, D the centre, and let 
AE be nearer the centre than AF, so that the 
perpendicular DG on AE is less than the perpen- 
dicular DIfoa AF. 

Then sqs. on DG, G£ = sq%. on DH, HF, 
and sqs. on DG, GA = sqs. on DH, HA. 

But sq. on I>G < sq. on DH. 




III. 7, 8] 



PROPOSITIONS 7, 8 



iJ 




Therefore sq. on GE > sq. on MJ^l 

iind sq. on GA > sq. on ffA, 

whence G£ > J/F, 

GA^HA. ,.- . 

Therefore, by addition, AE > AF. 
The other case taken by Heron is that where 
one perpendicular fails on the line produced, as in 
the annexed figure. In this case we prove in like 
manner that GE > HF, 

and GA > AH, 

Thus AE is greater than the sum of HF, AH, 
whence, a fortiori, AE is greater than the difference 
of HF, AH, i.e. than AF. 

Heron does not give the third possible case, that, namely, where both 
perpendiculars fall on the lines produced, The fact 
is that, in this case, the foregoing method breaks 
down. Though AE be nearer to the centre than 
AF'in the serjse that DG is less than DH, 
AE is not greater but less than AF. 
Moreover this cannot be proved by the same 
method as before. 

For, while we can prove that 

GE> HF, 
GA > AH, 
we ca.nnot make any inference as to the comparative length of AE, AF. 

To judge by Heron's corresponding note to in. 8, he would, to prove this 
case, practically prove iii. 35 first, i.e. prove that, if EA be produced to K 
and FA to Z, 

rect. FA, AL = lect. £A, AK, 
from which he would infer that, since AK^ AL by the first case, 

AE <AF. 
An excellent moral can, I think, be drawn from the note of Heroa 
Having the appearance of supplementing, or giving an alternative for, Euclid's 
proposition, it cannot be said to do more than confuse the subject. Nor was 
It necessary to find a new proof for the cast where the two lines which are 
compared are on epposiit sides of the diameter, since Euclid shows that for each 
line from the point to the circumference on one side of the diameter there is 
another of the same length equally inclined to it on the other side. 




\ 



Proposition 8. 

!f a point be taken outside a circle and front the point 
straight lines be drawn through to the circle, one of which 
is through the centre and the others are drawn at random, 
then, of the straight lines which fall on the concave circum- 
ference, that through the centre is greatest, while of the rest 



1» 



BOOK III 



[ill. 8 



ike nearer to that through the centre is always greater than 
the more remote, but, of the straight lines falling on the convex 
circumference, that between the point and the diameter is least, 
while of the rest the nearer to the least is always less than the 
more remote^ and only two equal straight lines will fall on the 
circle from the point, one on each side of the least. 

Let ABC be a circle, and let a point D be taken outside 
ABC; let there be drawn through 
from it straight lines DA, DE, DF, 
DC, and let DA be through the centre ; 
I say that, of the straight lines falling 
on the concave circumference AEFC, 
the straight line DA through the centre 
is greatest, 

while DE is greater than DF and DF 
than DC; 

but, of the straight lines falling on the 
convex circumference HLKG, the 
straight line DG between the point 
and the diameter AG is least; and 
the nearer to the least DG is always 
less than the more remote, namely DK 
than DL, and DL than DH. 

For let the centre of the circle ABC be taken [m. i], and 
let xth^M; let ME, MF, MC, MK, ML, MH be joined. 

Then, since AM is equal to EM, 
let MD be added to each ; 

therefore AD is equal to EM, MD. 

But EM, MD are greater than ED ; [i. lo] 

therefore AD is also greater than ED. 

Again, since ME is equal to MF, 

and MD is common, 
therefore EM, MD are equal to FM, MD ; 

and the angle EMD is greater than the angle FMD ; 

therefore the base ED is greater than the base FD. 

[..«4] 
Similarly we can prove that FD is greater than CD ; 

therefore DA is greatest, while DE is greater than DF, 

and DF than DC. 




III. 8] PROPOSITION 8 19 

Next, since MK, KD are greater than MD, [i. 30] 

and MG is equal to MK, 

therefore the remainder KD is greater than the remainder 
GD, 

so that GD is less than KD, 

And, since on MD, one of the sides of the triangle MLD, 
two straight lines MK, KD were constructed meeting within 
the triangle, 

therefore MK, KD are less than ML, LD \ ., [i. 21] 

and MK is equal to ML ; 

therefore the remainder DK is less than the remainder 
DL. 

Similarly we can prove that DL is also less than DH ; 
therefore DG is least, while DK is less than DL, and 
DL than DH. 

I say also that only two equal straight lines will fall from 
the point D on the circle, one on each side of the least DG, 

On the straight line MD, and at the point M on it, 
let the angle DMB be constructed equal to the angle KMD, 
and let DB be joined. 

Then, since MK is equal to MB, 
and MD is common, 

the two sides KM, MD are equal to the two sides BM, 
MD respectively ; 
and the angle KMD is equal to the angle BMD ; 

therefore the base DK is equal to the base DB. [i. 4] 

I say that no other straight line equal to the straight line 
DK will fall on the circle from the point D. 

For, if possible, let a straight line so fall, and let it be DN, 
Then, since DK is equal to DN, 

while DK is equal to DB, 

DB is also equal to DN, 
that is, the nearer to the least DG equal to the more remote: 
which was proved impossible. 

Therefore no more than two equal straight lines will fall 
on the circle ABC from the point D, one on each side ot 
ZJ£? the least. 

Therefore etc. 



90 



BOOK III 



[ill. 8 



As De Morgan points out, there are here two assumptions similar to 
that tacitly made in the proof of iii. 7, nameSy that 
K falls within the triangle DLM and E outside 
the triangle DFM. These facts can be proved 
in the same way as the assumption in iii. 7. Let 
DE meet FM in K and LM in Z Then, as 
before, MZ is less than ML and therefore than 
MK, Therefore K lies further than Z from 
the foot of the perpendicular from M on DE. 
Similarly E lies further than Y from the foot of the 
same perpendicular. 

Heron deals with lines on opposite sides of the 
diameter through the external point in a manner similar to that adopted in 
his previous note. 

For the case where E, F sk the seeond points in 
which AE, AF meet the circle the method answers 
well enough. 

If AE is nearer the centre D than AF is, 

sqs. on DG, GE = sqs. on DH, HF 
and sqs, on DG^ GA = sqs, on DH, HA, 




whence, since 
it follows that 

and 
so that, by addition. 




DG < DM, 
GE>HF, 
AG>AIf, 

AE > AF.- 
But, if ^, Z be the points in which AE, A F first 
meet the circle, the method fails, and Heron is reduced to proving, in the first 
instance, the property usually deduced from 111. 36. He argues thus : 
AKD being an obtuse angle, 
sq, on AD = sum of sqs. on AK, KD and twice rect AK, KG. [». 11] 
ALD is also an obtuse angle, and it follows that 

sum of sqs. on AK, KD and twice rect, AK, KG is equal to 

sum of sqs. on AL, LD and twice rect. AL, LB. '" 

Therefore, the squares on KD, LD being eqjal, 
sq on AK ATid. twice rect AK, KG = sq. on AL and twice rect. AL, LH, 
or sq on AKm\6 rect. AK, ^£ = sq. on AL and rect. AL, LF, 

i.e. rect. AK, AE = reci. AL, AF. 

But, by the first part, AE > AF. 

Therefore AK<AL. 

in. 7, 8 deal with the lengths of the several lines drawn to the circum- 
ference of a circle {1) from a point within it, {2) from a point outside it; but a 
similar proposition is true of straight lines drawn from a point on the 
circumference itself: If any point be taken on the circumference of a circle^ 
then, ofalltht straight lines which can be drawn from it to the circumference, the 
greatest is that in which the centre is ; of any others that which is nearer to the 
straight line which passes through the centre is greater than one more remote ; 
and from the same point there can be drawn to the circumference two straight 
lines, and only t^vo, which are equal fo one another, one on each side of the 
greatest line. 



>ai' 



III. 8, 9] PROPOSITIONS 8, 9 ai 

The converses of in, 7, 8 and of the proposition just given are also true 
and can easily be proved by reducHo ad ahurdum. They could be employed 
to throw light on such questions as that of internal contact, and the relative 
position of the centres of circles so touching. This is clear when part of the 
converses is stated : thus (i) if from any point in the plane of a circle a 
number of straight lines be drawn to the circumference of the circle, and one 
of these is greater than any other, the centre of the circle must lie on that one, 
{1) if one of them is less than any other, then, (a) if the point is within the 
circle^ the centre is on the minimum straight line produced i^emi the point, 
(i) if the point is outside the circle, the centre is on the minimum straight line 
prioduced btyond the point in which if meets the drck. 



Proposition 9. 

!/ a point be taken within a circle, and more than two 
equal straight lines fall from the point on the circle, the point 
taken is the centre of the circle. 

Let ABC be a circle and D a point within it, and from 
D let more than two equal straight 
lines, namely DA, DB, DC, fall on 
the circle ABC ; 

I say that the point D is the centre 
of the circle ABC. 

For let AB, BC be joined and 
bisected at the points B, F, and let 
ED, FD be joined and drawn through 
to the points G, K, H, L. 

Then, since AE is equal to EB, 
and ED is common, 

the two sides AF, ED are equal to the two sides BE, ED ; 

and the base DA is equal to the base DB ; 

therefore the angle AED is equal to the angle BED. 

[1.8] 
Therefore each of the angles AED, BED is right ; 

[i. Def. lo] 
therefore GK cuts AB into two equal parts and at right 
angles. 

And since, if in a circle a straight line cut a straight line 
into two equal parts and at right angles, the centre of the 
circle is on the cutting straight line, [ni, i. Pot.) 

the centre of the circle is on GK. 




si BOOK III [hi. 9 

For the same reason '■"" " ' ■ •'^ "^ * ■■ ' 

the centre of the circle ABC is also on HL. 

And the straight lines GK, HL have no other point 
common but the point D ; 

therefore the point D is the centre of the circle ABC. 

Therefore etc. Q. e. d. 

The result of this proposition is quoted by Aristotle, MettorolegUa in, 3, 
373 a 13 — 16 (cf, note on i. 8). 

III. 9 is, as De Morgan remarks, a loguai equivalent of part of in. 7, 
where it is proved that every (w>«-centra.l point is not a point from which three 
equal straight lines can be drawn to the circle. Thus 111. 7 says that every 
nht-A is not-B, and in. 9 states the equivalent fact that every B \% A. 
Mr H. M. Taylor does in effect make a logical inference of the theorem that, 
If from a point three equal straight linei tan be drawn (0 a circle^ that point is 
the centre, by making it a corollary to his proposition which includes the part of 
in. 7 referred to. Euclid does not allow himself these logical inferences, as we 
shall have occasion to observe elsewhere also. 

Of the two proofs of this proposition given in earlier texts of Euclid, 
August and Heiberg regard that translated above as genuine, relegating the 
other, which Simson gave alone, to a place in an Appendix. Camerer remarks 
that the genuine proof should also have contemplated the case in which one 
or other of the straight lines AB, BC passes through D. This would however 
have been a departure from Euclid's manner of taking the most obscure case 
for proof and leaving others to the reader. 

The other proof, that selected by Simson, is as follows : 

" For let a point D be taken within the circle ABC, and from D let more 
than two equal straight lines, namely AD, DB, DC, 
fall on the circle ABC ; 

I say that the point D so taken is the centrt: of the 
circle ABC. 

For suppose it is not ; but, if possible, let it be 
£, and let D£ be joined and carried through to the 
points J^, G. 

Therefore fV is a diameter of the circle ABC. 

Since, then, on the diameter FG of the circle 
ABC a point has been taken which is not the centre 
of the circle, namely D, 

DG n greatest, and DC is greater than DB, and DB than DA, . 

But the latter are also equal : which is impossible 

Therefore E is not the centre of the circle. 

Similarly we can prove that neither is any other point except D; '•' ■ '-' 
therefore the point D is the centre of the circle ABC. "^ ' 

.... r ..I Q. E. D." 

On this Todhunter correctly points out that the point E might be 
supposed to fall within the angle ADC. It cannot then be shown that DC 
is greater than DB and DB than DA, but only that either i?C or DA is [ess 
than DB ; this however is sufficient for establishing the proposition. 





III. lo] PRur-OSlTIONS 9, lo t3 

Proposition io, 

A circle does not cut a circle at more points than two. 

For, if possible, let the circle ABC cut the circle DBF 
at more points than two, namely 
B, C, /*, Ii \ 

let BH, BG be joined and 
bisected at the points K, L, 
and from K, L let KC, LM be 
drawn at right angles to BH, 
BG and carried through to the 
points A, E. 

Then, since in the circle 
ABC a straight line AC cuts a 
straight line BH into two equal 
parts and at right angles, 

the centre of the circle ABC is on AC. [in- i, For.] 

Again, since in the same circle ABC a straight line NO 
cuts a straight line BG into two equal parts and at right 
angles, 

the centre of the circle ABC is on NO. 

But it was also proved to be on AC, and the straight 
lines AC, NO meet at no point except at P ; 

therefore the point P is the centre of the circle ABC. 

Similarly we can prove that P is also the centre of the 
circle DEF\ 

therefore the two circles ABC, DBF which cut one 
another have the same centre P : which is impossible, [in- s] 

Therefore etc. q. e. d. 

I. The won) circle (niiXii)) ii here employed in the uousual (Case of the eireum/tremt 
{npt^ptM] of « drck. Cf. note on i. Der. ii,. 

There is nothing in the demonstration of this proposition which assumes 
that the circles cul one another ; it proves that two circles cannot mtet at mor^ 
than two points, whether they cut or meet without cutting, i.e. iouch one 
another, 

Hete again, of two demonstrations given in the earlier texts, Simson chos« 
the second, which Au(;u3t and Keilicrg relegate to an Appendix and which is 
as follows : 

" For again let the circle ABC cut the circle DEF at more points than 
two, namely B, G, H, F\ 

let the centre K of the circle ABC be taken, and let KB, KG^ KF be 
joined. 



BOOK III 



[hi, IO, II 




Since then a point K has been taken within the circle DEF, 
and from K more than two straight lines, namely 
KB, KF, KG, have fallen on the circle DEF, 
the point A' is the centre of the circle DEF. [in. 9] 

But K is also the centre of the circle ABC. 

Therefore two circles cutting one another have 
the same centre K : which is impossible, [111. 5] 

Therefore a circle does not cut a circle at more 
points than two. 

Q. E. D." 

This demonstration is claimed by Heron (see an-NairizI, ed, Curtie, 
pp. I JO — i). It is incomplete because it assumes that the point K which is 
taken as the centre of the circle ABC is within the circle DEF. It can 
however be completed by means of hi. 8 and the corresponding proposition 
with reference to a point on the circumference of a circle which was enunciated 
in the note on m. 8. For (i) if the point K is en the circumference of the 
circle DEF, we obtain a contradiction of the latter proposition which asserts 
that only two equal straight lines can be drawn from K to the circumference 
of the circle DEF; (i) if the point K is outside the circle DEF, we obtain a 
contradiction of the corresponding part of [ii. 8. 

Euclid's proof contains an unproved assumption, namely that the lines 
bisecting BG, BH at right angles will meet in a point P. For a discussion 
of this assumption see note on ir. 5. 



Proposition i i, 

If tivo circles touch one another internally, and their centres 
be taken, the straight line joining their centres, if it be also 
produced, will fall on the point of contact of the circles. 

For let the two circles ABC, ADE touch one another 
internally at the point A, and lei 
the centre F of the circle ABC, and 
the centre G of ADE, be taken ; 
I say that the straight line joined 
from G Xo F and produced will fall 
on A. 

For suppose it does not, but, 
if possible, let it fall hs, FGH, and 
let A F, AG he joined. 

Then, since J^G, 6"/^ are greater 
than FA, that is, than FH, 

let FG be subtracted from each ; 

therefore the remainder AG is greater than the remainder 
GH. 




HI. iij PROPOSITIONS lo, u »S 

But AG is equal to GD ; 
therefore GD is also greater than G//, 
the less than the greater : which is impossible. 
Therefore the straight line joined from F to G will not 
fall outside ; 

therefore it will fall at A on the point of contact. 
Therefore etc, 

Q. E. D. 

i. the straight line joining their centres, literally "the straight line joined to their 
3. point of contact is here trtira^, and in the enunciation ur the next proposition 

Again August and Heiberg give in an Appendix the additional or 
alternative proof, which however shows little or no variation from the genuine 
proof and can therefore well be dispensed with. 

The genuine proof is beset with difficulties in consequence of what tt 
tacitly assumes in the figure, on the ground, probably, of its being obvious to 
the eye, Camerer has set out these difficulties in a most careful ^ote, the 
heads of which tnay be given as follows : 

He observes, first, that the straight line joining the centres, when produced, 
must necessarily (though this is not stated by Euclid) he produced in the 
dirtdion of the centre of the circle which touches the ether inltrnally. (For 
brevity, I shall call this circle the " inner circle," though I shall imply nothing 
by that term except thai it is the circle which touches the other on the inner 
side of the latter, and therefore that, in accordance with the definition of 
touching, points on it in the immediate neighbourhood of the point of contact 
are necessarily within the circle which it touches.) Camerer then proceeds by 
the following steps. 

T. The two circles, touching at the given point, cannot intersect at any 
f>oint. For, since points on the "inner" in the immediate neighbourhood of 
the point of contact are within the "outer" circle, the inner circle, if it 
intersects the other anywhere, must pass outside it and then return. This is 
only possible (o) if it passes out at one point and returns at another point, or 
{b) if it passes out and returns through one and the same point (a) is impossible 
because it would require two circles to have three common points ; {i) would 
require that the inner circle should have a node at the point where it passes 
outside the other, and this is proved to be impossible by drawing any radius 
cutting both loops. 

*. Since the circles cannot intersect, one must be entire^ within the 
other. 

3, Therefore the outer circle must be greater than the inner, and the 
radius of the outer greater than that of the inner. 

4. Now, if /■ be the centre of the greater and G of the inner circle, and 
if FG produced beyond G does not pass throt^h A, the given point of 
contact, then there are three possible hypotheses. ;. , 

(a) A may lie on GF produced beyond F. ' ■ ■ . 



36 BOOK IIT [lit. II 

(#) A may lie outside the line FG altogether, in which case J^G produced 

beyond G must, in consequence of result 3 above, either 
(i) meet the circles in a point common to both, or 
{ii) meet the cirdes in two points, of which that which is on the inner 

circle is nearer to G than the other is. 

(a) is then proved to be impossible by means of the fact that the radius of the 

inner circle is less than the radius of the outer. 

{&) (ii) is Euclid's case ; and his proof holds equally of {#) (i), the hypothois, 

namely, that £> and Jf in the figure coincide- 
Thus all alternative hypotheses are successively shown to be impossible, 

and the proposition is completely established. 

I think, however, that this procedure may be somewhat shortened in the 
following manner. 

In order to make Euclid's proof absolutely conclusive we have only (i) to 
take care to produce /^G beyond G, the centre of the " inner " circle, and then 
(a) to prove that the point in which J^G so produced meets the *' inner " circle 
is nai further from G than is the point in which it meets the other circle. 
Euclid's proof is equally valid whether the first point is nearer to G than the 
second or the first point and the second coincide. 

If FG produced beyond G does not pass through A, there are two 





conceivable hypotheses : (a) A may lie on GF produced beyond F, or (i) A 
may be outside FG produced either way. In either case, if FG produced 
meets the " inner " circle in D and the other in H, and if GD is greater than 
GH, then the " inner " circle must cut the " outer " circle at some point 
between A and D, say X. 

But if two circles have a common point X lying on one side of the line of 
centres, they must have another conesponding point on the other side of the 
line of centres. This is clear from in. 7, 8 ; for the point is determined by 
drawing from F and G, on the opposite side to that where X is, straight 
lines FY, G Y making with FD angles equal to the angles DFX, DGX 
respectively. 

Hence the two circles will have at least three points common : which is 
impossible. 

Therefore GD cannot be greater than GH; accordingly GD must be 
either equal to, or less than, GH, and Euclid's proof is valid. 

The particular hypothesis in which FG is supposed to be in the same 
straight line with A but G is on the side of F^way from A is easily disposed 
of, and would in any case have been left to the reader by Euclid. 

For GD is either equal to or less than GH. 

Therefore GD is less than Fff, and therefore less than FA, 

But GD is equal to GA, and therefore greater than FA : which is 
impossible. 




in. II, i2j PROPOSITIONS n, la ay 

Subject to the same preliminary investigation as that required by Euclid's 
proof, the proposition can also be proved directly from iii. 7. 

For, by iii. 7, GH\% the shortest straight line that can be drawn from G 
to the circle with centre F; 

therefore GH\s less than GA, ' ' 

and therefore less than GD : which is absurd. 

This proposition is the crucial one as regards circles which touch internally; 
and, when it is once established, the relative position of the circles can be 
completely elucidated by means of it and the propositions which have preceded 
it. Thus, in the annexed f^ure, if ^ be the centre 
of the outer circle and G the centre of the inner, 
and if any radius FQ of the outer circle meet the 
two circles in Q, P respectively, it follows, from 
III. 7, in. 8, or the corresponding theorem with 
reference to a point on the circumference, that FA 
is the maximum straight line from .^to the circum- 
ference of the inner circle, FP is less than FA, 
and FP diminishes in length as FQ moves round 
from FA until FP reaches its minimum length 
FB. Hence the circles do not meet at any other 
point than A, and the distance PQ cut off between them on any radius FQ 
of the outer circle becomes greater and greater as FQ_ moves round from FA 
to FC and is a maximum when FQ coincides with FC, after which it 
diminishes again on the other side of FC. 

The same consideration gives the partial converse of in. 11 which forms 
the 6th lemma of Pappus to the first book of the Tactioms of Apollonius 
(Pappus, vn. p. 826). This is to the effect that, if h'&, AC art in ont straight 
lim, and on ont side of A, tht cirda described on AB, AC as diameters touch 
(internally at the point A). Pappus concludes this from the fact that the 
circles have a common tangent at A ; but the truth of it is clear from the fact 
that FP diminishes as FQ moves away from FA on either side ; whence the 
circles meet at A hut do not cut one another. 

Pappus' 5th lemma (vn. p. 824) is another partial converse, namely that, 
pven two circles touching internally at A, and a lint ABC drawn from A cutting 
both, then, if the centre of the outer circle lies on ABC, so does the centre -of the 
inner. Pappus himself proves this, by means of the common tangent to the 
circles at A, in two ways, (i) The tangent is at right angles to .^C and 
therefore to AB'. therefore the centre Qi the inner circle lies on AB. (2) By 
in. 32, the angles in the alternate segnients of both circles are right angles, so 
that ABC is a diameter of both. 

[Proposition 12. 

If two circles touch one another externally, the straight 
line joining their centres will pass through the point of 
contact. 

For let the two circles ABC, ADE touch one another 
S externally at the point A, and let the centre Foi ABC, and 
the centre G of ADE, be taken ; 




38 - BOOK III [i[i. It 

I say that the straight line joined from F to G will pass 
through the point of contact at A. 
For suppose it does not, 
'" but, if possible, let it pass as 
FCDG, and let AF, AG be 
joined. 

Then, since the point F is 
the centre of the circle ABC, 
IS FA is equal to FC. 

Again, since the point G is 
the centre of the circle ADE, 
GA is equal to GD. 
But FA was also proved equal to FC ; 
» therefore FA, AG are equal to FC, GD, 

so that the whole FG is greater than FA, AG ; 
but it is also less [i. zo] : which is impossible, 

Therefore the straight line joined from F to G will not 
fail to pass through the point of contact at A ; 
n therefore it will pass through it. 

Therefore etc. >< • ' q, e. D.j 

ij. win not fall lo pang. The Greek has the doubk negaliye, o(ic dpe V-.'Wwli... 
adK iXedrirai, Literally ^' the straight line... will not ttef-^as,,.," 

Heron says on iii, 1 1 : " Euclid in proposition 1 1 has suppostKl the two 
circles to touch internally, made his proposition deal with this case and proved 
what was sought in it, Buf I will show how if is to be proved if the contact is 
external." He then gives substantially the proof and figure of ill, i». It 
seems clear that neither Heron nor an-Nairiit had ni. 1 2 in this place, 

Campanus and the Arabic edition of Naslraddin at-fQsI have nothing more 
of III. 12 than the following addition to 111. 11. "In the case of external 
contact the two lines ae and eb will be greater than ai, whence ad and cb will 
be greater than the whole ai, which is false." (The points a, b, c, d, e cor- 
respond respectively to G, P, C, D, A in the above figure.) It is most 
probable that Theon or some other editor added Heron's prt>of in his edition 
and made Prop. 12 out of it {an-Nairlit, ed, Curtze, pp. 121 — 2). An-NairM 
and Campanus, conformably with what has been said, number Prop. 13 of 
Hei berg's text Prop, i z, and so on through the Book. 

What was said in the note on the last proposition applies, mutatis mutandis, 
to this, Camerer proceeds in the same manner as before ; and we may use 
the same alternative argument in this case also. 

Euclid's proof is valid provided only that, if FG, joining the assumed 
centres, meets the circle with centre F in C and the other circle in D, C is 
not within the circle ADE and D is not within the circle ABC. {The proof 
is equally valid whether C, D coincide or the successive points are, as drawn 
in the figure, in the order F, C, D, G.) Now, if C is within the circle ADE 




III. I a] PROPOSITION ii 39 

and D within the circle ABC, the circles must have cut between A and C 
and between A and D. Hence, as before, they must also have another 
corresponding point common on the other side of CO. That is, the circles 
must have three common points : which is impossible. 

Hence Euclid's proof is valid W F, A, G form a triangle, and the only 
hypothesis which has still to be disproved is the 
hypothesis which he would in any case have left to 
the reader, namely that A does not lie on FG but 
on FG produced in either direction. In this case, as 
before, either C, D must coincide or C is nearer 
/"than D is. Then the radius FC must be equal 
to FA : which is impossible, since FC cannot be 
greater than FD, and must therefore be less than 
FA. 

Given the same preliminaries, in, u can be proved by means of 111. 8, 

Again, when the proposition in. 12 is once proved, in, S helps us to prove 
at once that the circles He entirely outside each other and have no other 
common point than the point of contact. 

Among Pappus' lemmas to Aptollonius' Tactiones are the two partial 
converses of this proposition corresponding to those given in the last note. 
Lemma 4 (vii. p. 824) is to the effect that, tf AB, AC be in one straight tint, B 
and C bang on opposite sides 0/ A, the circles drawn on AB, AC as diameters 
touih externally at A. Lemma 3 (vii. p. 822) states that, 1/ two circles touch 
externally at A and BAC is drawn through h cutting both circles and containing 
the centre of one, BAC will also contain the centre of the other. The proofs, as 
before, use the common tangent at A. 

Mr H. M, Taylor gets over the difficulties involved by in. 11, 12 in a 
manner which is most ingenious but not Euclidean. He first proves that, j^rtco 
circles meet at a point not in the same straight line with their centres, the circles 
intersect at thai point ; this is very easily established by means of in. 7, 8 and 
the third similar theorem. Then he gives as a corollary the statement that, if 
two circles toueh, the point of contact is in the same straight line with their 
centres. It is not explained how this is inferred from the substantive 
proposition ; it seems, however, to be a logical inference simply. By the 
proposition, every A (circles meeting at a point not in the same straight line 
with the centre) is B (circles which intersect); therefore every not-^ is not' A, 
i.e. circles which do not intersect do not meet at a point not in the same 
straight line with the centres. Now non-intersecting circles may either meet 
{i.e. touch) or not meet. In the former case they must meet en the line of 
centres ; for, if they met at a point not in that line, they would intersect. But 
such a purely logical inference is foreign to Euclid's manner. As De Moi^an 
says, *' Euclid may have been ignorant of the identity of ' Every X is Y' and 
' Every not- Y is noi-X,' for anything that appears in his writings ; he makes 
the one follow from the other by a new proof each time " (quoted in Keynes' 
Formal Legie, p. 8r), 

There is no difficulty in proving, by means of i. 20, Mr Taylor's next 
profMJsition that, if two circles meet at a point which lies in the same straight 
line as their centres and is between the centres, the circles touch at that point, and 
each circle lies without the ether. But the similar proof, by means of e. so, of 
the corresponding theorem for internal contact seems to be open to the same 
objection as Euclid's proof of in. 11 in that it assumes without proof that the 
circle which has its centre nearest to the point of meeting is the "inner" 
circle. Lastly, in order to prove that, if two circles hm'e a point of contact, they 



30 



BOOK III 



[ill. i> 



do not mat at any other point, Mr Taylor uses the qtiestionable corollary. 
Therefore in any case his alternative procedure doet not seem prefer&ble to 
Euclid's. 

The altcjrnative to Eucl. HI. ii — 13 which finds most favour in modem 
continental text-books (e.g. L^endre, Baltzer, Henrici and Treutlein, 
Veronese, Ingrami, Enriques and Amaldi) connects the number, position and 
nature of the coincidences between points on two circles with the relation in 
which the distance between their centres stands to the length of their radii. 
Enriqties and Amaldi, whose treatment of the different cases is typical, give 
the following propositions (Veronese gives them in the converse form). 

I . If the distance between the centres of two circles is greater than the stint 
of the radii, the two circles have no point common and are external to one 
another. 

Let O, ff be the centres of the circles (which we will call " the circles 
0, O "), r, r their radii respectively. 

Since then OO >r-\- r', a fortiori OO ->r, and O is therefore exterior to 
the circle O. 

Next, the circumference of the circle intersects OG in a point A, and 
since 0O>r-¥r\ AO>r', and A is 
external to the circle O. 

But (7A is less than any straight 
line, as OB, drawn to the circum- 
ference of the circle O [in. 8] ; hence 
all points, as B, on the circumference 
of the circle are external to the circle 

^- . . \ yo 

Lastly, if C be any point internal 

to the circle 0, the sum of (7C, f^C is ' 

greater than (/O, and a fortiori grtaX&r than r-\-r'. 

But OC is less than r: therefore OC is greater than t', or C is external 
to 0. 

Similarly we prove that any point on or within the circumference of the 
circle O is external to the circle 0- 

a. If the distance between the centres of two unequal circles is less than the 
difference of the radii, the two circumferences have no eontnton point and the lesser 
circle is entirety within the greater. 

Let 0, C be the centres of the two circles, r, r' their radii respectively 

Since Off <.r — r, a fortiori Off < r*, so that is 
internal to the circle O. 

If A, A' be the points in which the straight line 
00 intersects respectively the circumferences of the 
circles 0, O, 

00 is less than ffA'-OA, 
so that (7(3 + OA, or ffA, is less than OA', 
and therefore A is internal to the circle ff. 

But, of all the straight lines from O to the circumference of the circle O, 
OA passing through the centre O is the greatest [in, 7] ; 
whence all the points of the circumference of are internal to the circle O. 

A similar argument to the preceding will show that all points within the 
circle O are internal to the circle O. 





III. is] 



PROPOSITION 13 



31 




3. If the diitance btilvan the centra of two cirdti is equal to tht tuM of Iht 

radii, tht two drcumfirtnces have one point (ommon and one onfy, and that point 
is on the Um of an f res. Each circle is externa/ to the other. 

Let O, C be the centres, r, r the radii of the circles, so that OO \i equal 
\ar*-r'. 

Thus 00 is greater than r, so that O 
is external to the circle O, and the circum- 
ference of the circle O cuts OO in a 
point A. 

And, since OO is equal to /■ + r*, and 
OA to r, it follows that 0A is equal to r\ 
so that A belongs also to the circumference 
of the circle O. 

The proof that all other points on, and 
all points within, the circumference of the circle O are external to the cincle O 
follows the similar proof of prop. 1 above. And similarly all points (except A) 
on, and all points within, the circumference of the circle O are external to the 
circle O. 

The two circles, having one common point only, touch at that point, which 
lies, as shown, on the line of centres. And, since the circles are external to 
one another, they touch externally. 

4. If the distance between the centres of two unequal circles is equal to tht 
difference between the radii, the two circumferences have one point and one only in 
common, and that point lies on the line of centres. The lesser circle it within tha 
other. 

The proof is that of prop. 2 above, mutatis mutandis. 

The circles here touch internally at the point on the line of centres. 

5. If the distance between the centres of two circles is less than the sum, and 
greater than the di^erence, of the radii, the two circumferences hive two common 
points symmetrically situated with respect to the line of centres but not lying on 
that line. 

Let O, O \x the centres of the two circles, r, r their radii, *' being the 
greater, so that 

r'-r<Oa <r + *'. 

It follows that in any case 00 + /•> r', so that, if DM be taken on ffO 
produced equal to r (so that M is on the circumference of the circle 0), At is 
external to the circle ff.' 

We have to use the same Postulate as in Eucl. I, 1 that 

An arc of a circle which has one extremity within and the other without a 
given circle has one point common with the 
latter and only one ; from which it follows, 
if we consider two such arcs making a 
complete circumference, that, if a circum- 
ference of a circle passes through one point 
internal to, and one point external to a 
given circle, it cuts the latter circle in two 
points. 

We have then to prove that the circle O, 
besides having one point M of its circum- 
ference external to the circle ff, has one other point of its circumference (Z) 
internal to the latter circle. 




3« 



BOOK in 



[ill. 11, 13 




Three cases have to be distinguished according as 0^7 is greater than, equal 
to, or less than, the radius r of the lesser circle. 

(1) 00' > r, (See the preceding figure.) 

Measure OL along Off equal to r, so that 
Z lies on the circumference of the circle O. 

Then, since Off < r + ^, OL will be less 
than r, so that L is within the circle ff. 

{2) Off^r. 

In this case the circumference of the circle 
passes through ff, or L coincides with ff. 

(3) Off<^r. 

If we measure OL along Off equal to r, the point L will lie on the 
circumference of the circle O. 

Then OL^r- Off, 
so that O'L < r, and a fortiori ffL < r\ so that Z 
lies within the circle ff. 

Thus, in all three cases, since the circumference 
of O passes through one point {M) external to, and 
one point (L) internal to, the circle ff, the two 
circumferences intersect in two points A, B [Post. J 

And A, B cannot lie on the line of centres OO, 
since this straight tine intersects the circle O in 
L, M only, and of these points one is inside, the other outside, the circle O. 

Since AB\^a. common chord of both circles, the straight line bisecting it 
at right angles passes through both centres, i.e. is identical with Off ■ 

And again by means of 111. 7, 8 we prove that all points except A, B on 
the arc ALB lie within the circle ff, and all points except A^ B on the arc 
A MB outside that circle ; and so on. 




Proposition 13, 

A circle does not touch a circle at more points t/ian one, 
whether it touch it internally or externally. 

For, if possible, let the circle ABDC touch the circle 
EBFD, first internally, at more 
5 points than one, namely D, B. 
Let the centre G of the circle 
ABDC, and the centre H of 
EBFD, be taken. 

Therefore the straight line 
10 joined from G to /f will fall on 
B, D. [in. 11] 

Let it so fall, as BGHD. 
Then, since the point G is 
the centre of the circle A BCD, 
IS BG is equal to GD ; 




'^ m. 13] PROPOSITIONS ii, 13 3$ 

therefore BG is greater than ZfD ; » -- ' 

therefore B// is much greater than HD. 
Again, since the point // is the centre of the circle 
EBFD, 
ao B// is equal to //D ; 

but it was also proved much greater than it : which is 
impossible. 

Therefore a circle does not touch a circle internally at 
more points than one. .^^ 

*s I say further that neither does it so touch it externally. 
For, if possible, let the circle ACK touch the circle 
ABDC at more points than one, namely A, C, 
and let AC be joined. 

Then, since on the circumference of each of the circles 

JO ABDC, A CK two points A , C have been taken at random, 

the straight line joining the points will fall within each 

, circle ; [m. 

but it fell within the circle ABCD and outside ACK 

[ni. Def, 3] : which is absurd. 

js Therefore a circle does not touch a circle externally at 
more points than one. 

And it was proved that neither does it so touch it 
internally. 

Therefore etc. Q. e. d. 

3, 7, [4, 37, 30. 33. ABDC Euclid writes ABCD (hew and in the next proposition), 
notwithstiuiding the order in which the points are placed in the iieure. 

tj, iT- does it so touch it. It is necessary to supply these words which the Gr«ek 
(Dri uliti licTin and fri oiSi itTij) leaves to be understood. 

The difficulties which have been felt in regard to the proofs of this 
proposition need not trouble us now, because they have already been disposed 
gf in the discussion of the more crucial propositions in. 1 1, 11. 

Euclid's proof of the first part of the proposition differs from Simson's ; 
and we will deal with Euclid's first. On this Cannerer remarks that it is 
assumed that the supposed second point of contact lies on the line of centres 
productd beyond the centre of tht "outer" circle, whereas all that is proved in 
III, 1 1 is that the line of centres produced beyond the centre of the " inner'" circle 
passes through a point of contact. But, by the same argument as that given 
on ni, 11, we show that the circles cannot have a point of contact, or even 
any common point, outside the line of centres, because, if there were such a 
point, there would be a corresponding common point on the other side of the 
line, and the circles would have three common points. Hence the only 
hypothesis left is that the second point of contact may be on the line of 
centres but in the direction of the centre of the "tfw^^r" circle; and Euclid's 
proof disposes of thb hypothesis. 




34 BOOK in [ill. 13, 14 

Heron (in an-Nairlzi, ed. Curtze, pp. m — 4), curiously enough, does not 
question Euclid's assumption chat the line of centres passes through both 
points of contact (if double contact is possible) ; but he devotes some space to 
proving that the centre of the "outer" circle must lie within the "inner" circle, a 
fact which he represents Euclid as asserting (" sicut dixit Euclides "), though 
there is no such assertion in our text. The proof of the fact is of course easy. 
If the line of centres passes through fe/A points of contact, and the centre of 
the "outer" circle lies either on or outside the "inner" circle, the line of 
centres must cut the "inner" circle in /hrei points in all: which is impossible, 
as Heron shows by the lemma, which he places here (and proves by i. 16), 
that a straight line cannot cut the circumfertnct of a circle in mere points 
than two. 

Simson's proof is as follows (there is no real need for giving two figures as 
he does). 

" If it be possible, let the circle EBF touch the circle ABC in more 
points than one, and first on the inside, in the 
points B, J?; join BD, and draw G/f bisecting 
B£> at right angles. 

Therefore, because the points B, D are in the 
circumference of each of the circles, the straight 
line BD falls within each of them : And their 
centres are in the straight line GH which bisects 
BD at right angles : 

Therefore GH passes through the Doint of 
ccMitact [ill. I ij ; but it does not pass through it, 
because the points B, D are without the straight line GH: which is absurd. 

Therefore one circle cannot touch another on the inside in more points 
than one." 

On this Camerer remarks that, unless ill. 11 be more completely elucidated 
than it is by Euclid's demonstration, which Slmson has, it is not sufficiently 
clear that, besides the point of contact in which GH meets the circles, they 
cannot have another point of contact either (1) on GH or (i) outside it. 
Here again the latter supposition (2) is rendered im possible because in that 
case there would be a third common point on the opposite side of GH ; and 
the former supf)osition,(i) is that which Euclid's proof destroys. 

Simson retains Euclid's proof of the second part of the proposition, though 
his own proof of the first part would apply to the second part also if a 
reference to iii. 12 were substituted for the reference to in. 11. Euclid might 
also have proved the second part by the same method as that which he 
employs for the first part. 

Proposition 14. 

In a circle equal straight lines are equally distant from 
the centre, and those which are equally distant from the centre 
are equal to one another. 

Let ABDC be a circle, and let ABt CD be equal straight 
lines in it ; 

I say that AB, CD are equally distant from the centre, 

For let the centre of the circle ABDC be taken [jh. \\ 




III. 14] PROPOSITIONS 13, 14 3^5 

and let it be E\ from E let EF, EG be drawn perpendicular 
to AB, CD, and let AE, EC be joined. 

Then, since a straight line EF through 
the centre cuts a straight line AB not through 
the centre at right angles, it also bisects it. 

Therefore AF is equal to FB ; 
I therefore AB is double of AF, 

For the same reason 
I CD is also double of CG ; 

and AB is equal to CD ; 

therefore AF is also equal to CG. 
' And, since AE is equal to EC, 

the square oxs. AE\^ also equal to the square on EC. 

But the squares on AF, EF^r^ equal to the square on AE, 
for the angle at F is right ; 

and the squares on EG, GC are equal to the square on EC, 
for the angle at G is right ; [r. 47] 

therefore the squares on AF, FE are equal to the 
squares on CG, GE, 

of which the square on AF is equal to the square on CG, 
for AFis equal to CG ; 

therefore the square on FE which remains is equal to 
the square on EG, 

therefore EF is equal to EG 
But in a circle straight lines are said to be equally distant 
from the centre when the perpendiculars drawn to them from 
the centre are equal ; [ni. Def. 4] 

therefore AB, CD are equally distant from the centre. 

Next, let the straight hnes AB, CD be equally distant 
from the centre ; that is, let EF be equal to EG. 

I say that AB is also equal to CD. ~ ' ' 

For, with the same construction, we can prove, similarly, 
that AB is double of AF, and CD of CG. 

And, since AE is equal to CE, 

the square on AE is equal to the square on CE, 
But the squares on EF, FA are equal to the square on AE, 
and the squares on EG, GC equal to the square on CE. [i. 47] 



^ BOOK in - -"^ [ill. 14, »5 

Therefore the squares on £/^, FA are equal to the 
squares on £G, GC, i ■■ 

of which the square on EF is equal to the square on EG, 
for EF is equal to EG \ 

therefore the square on AF which remains is equal to the 
square on CG ; , , \ 

therefore ^^ is equal to CC " 

And y4^ is double of AF, and CD double of CG ; 

therefore AB is equal to CD. 
Therefore etc. 

Q. E. D. 

Heron (an-NairixI, pp. 125 — 7) has an elaborate addition to this proposition 
in which he proves, first by redtuiio ad aiiurdum, and then directly, that the 
centre of the circle falls between the two chords. 



Proposition 15. r,.. ... ,. • ..; 

0/ straight lines in a circle the diameter is greatest, 
and of the rest the nearer to the centre is always greater than 
the more remote. 

Let ABCD be a circle, let AD be its diameter and E 
the centre ; and let BC be nearer to the , , 
diameter AD, and FG more remote ; 
I say that AD is greatest and BC 
greater than FG. 

For from the centre E let EH, EK 
be drawn perpendicular to BC, FG. 

Then, since BC is nearer to the 
centre and FG more remote, EK is 
greater than EH. [in. Def. 5] 

Let EL be made equal to EH, 
through L let LM be drawn at right 
angles to EK and carried through to N, and let ME, EN, 
FE, EG be joined. 

Then, since EH is equal to EL, 

BC is also equd to MN. [m. 14] 

Again, since AE is equal to EM, and ED to EN, 
AD is equal to ME, EN. 




III. IS, 16} PROPOSITIONS 14—16 37 

But ME, EN are greater than MN, [1. ao] 

and MN is equal to BC\ ' ' 

therefore AD is greater than BC. 

And, since the two sides ME, EN are equal to the two 
sides FE, EG, 

and the angle MEN greater than the angle FEG, 

therefore the base MN is greater than the base FG, [i. 14] 

But MN was proved equal to BC. 

Therefore the diameter AD is greatest and BC greater 
than FG. 

Theretore etc. g. e. d, 

1. Of straight lines. The Greek leaves these words to be understood. 
5. Nearer to the diameter AD, As BC, FG are not In general parallel to AD, 
Euclid should have said ^' nearer to the centre." 

It will be observed that Euclid's proof differs from that given in our text- 
books (which is Simson's) in that Euclid introduces another line MN, which 
is drawn so as to be equal to BC but at right angles to EK and therefore 
parallel to FG. Sim son dispenses with MN^vA tases his proof on a similar 
proof by Theodosius {Spkatrica i, 6). He proves that the sum of the squares 
on EH, HB is equal to the sum of the squares on EK, KF\ whence he 
infers that, since the square on EH'\% less than the square on EK, the square 
on BH is greater than the square on FK. It may be that Euclid would have 
regarded this as too complicated an inference to make without explanation or 
without an increase in the number of his axioms. But, on the other hand, 
Euclid himself assumes that the angle subtended at the centre by MN is 
greater than the angle subtended by FG, or, in other words, that M, N both 
fall outside the triangle FEG. This is a similar assumption to that made in 
lit, 7, 8, as already noticed; and its truth is obvious because EM, EN, being 
r<idii of the circle, are greater than the distances from E to the points in which 
MN cuts EF, EG, and therefore the latter points are nearer than M, N&it to 
Z, the foot of the perpendicular from E to MN. 

Simson adds the converse of the proposition, proving it in the same way 
as he proves the proposition itself. 



Proposition i6. .r ••'•:, 

TAe slraighl line dragon ai right angles to the diameter 
of a circle from its extremity will fall outside the circle, and 
into the space between the straight line and the circumference 
another straight line cannot be interposed ; further the angle 
of the semicircle is greater, and the remaining angle less, than 
any acute rectilineal angle. 

Let ABC be a circle about D as centre and AB as 
diameter ; 



3S 



BOOK in 



fill. i6 




I say that the straight line drawn from A at right angles 
to AB from its extremity will fall -^ , 
outside the circle. 

For suppose it does not, but, 
if possible, let it fall within as CA, 
and let DC be joined. 

Since DA is equal to DC, 

the angle DAC is also equal to 
the angle A CD, [i. s] 

But the angle DAC is right ; 
therefore the angle ACD is also right : 
thus, in the triangle ACD, the two angles DAC, ACD are 
equal to two right angles : which is impossible. [i. 1 7] 

Therefore the straight line drawn from the point A at 
right angles to BA will not fall within the circle. 

Similarly we can prove that neither will it fall on the 
circumference ; 

therefore it will fall outside. 

Let it fall as AE ; 
I say next that into the space between the straight line AE 
and the circumference CHA another straight line cannot be 
interposed. 

For, if possible, let another straight line be so interposed, 
as EA, and let DG be drawn from the point D perpendicular 
to EA. -, 

Then, since the anrie A GD is right, ' 

and the angle DA G is less than a right angle, 

AD is greater than DG. [i. 19] 

But DA is equal to DH ; 

therefore DH is greater than DG, the less than the 
greater : which is impossible. 

Therefore another straight line cannot be interposed into 
the space between the straight line and the circumference, 

I say further that the angle of the semicircle contained by 
the straight line BA and the circumference CHA is greater 
than any acute rectilineal angle, 

and the remaining angle contained by the circumference CHA 

and the straight line AE is less than any acute rectilineal angle. 

For, if there is any rectilineal angle greater than the 

angle contained by the straight line BA and the circumference 



III. 16] PROPOSITION i6 39 

CHA, and any rectilineal angle less than the angle contained 
by the circumference CHA and the straight line AE, then 
into the space between the circumference and the straight line 
AE a straight line will be interposed such as will make an 
angle contained by straight lines which is greater than the 
angle contained by the straight line BA and the circumference 
CHA, and another angle contained by straight lines which 
is less than the angle contained by the circumference CHA 
and the straight line AE. 

But such a straight line cannot be interposed ; 

therefore there will not be any acute angle contained by 
straight lines which is greater than the angle contained by 
the straight line BA and the circumference CHA, nor yet 
any acute angle contained by straight lines which is less than 
the angle contained by the circumference CHA and the 
straight line AE. — 

PoKiSM. ' From this it is manifest that the straight line 
drawn at right angles to the diameter of a circle from its 
extremity touches tne circle. 

,. cannot be Interposed, Ut«ir>lly " will not fall in between" (od nptikwaCrat). 

This proposition is historically interesting because of the controversies to 
which the last part of it gave rise from the 13th to the 17th centuries. 
History was here repeating itself, for it is certain that, in ancient Greece, both 
before and after Euclid's time, there had been a great deal of the same sort 
of contention about the nature of the " angle of a semicircle " and the 
"remaining angle" between the circumference of the semicircle and the 
tangent at its extremity. As we have seen (note on i. Def. 8), the latter angle 
had a recognised name, iMparoitSij? yuivii, hern-tike or eomitu!ar angle ; 
though this term does not appear in Euclid, it is often used by Proctus, 
evidently as a term well understood. While it is from Proclus that we get the 
best idea of the ancient controversies on this subject, we may, I thinl^ infer 
their prevalence in Euclid's time from this solitary appearance of the two 
" angles " in the Elements. Along with the definition of the angle 0/ a 
segment, it seemi. to show that, although these angles are only mentioned to 
be dropped again immediately, and are of no use in elementary geometry, or 
even at all, Euclid thought that an allusion to them would be expected of 
him ; it is as if he merely meant to guard himself against appearing to ignore 
a subject which the geometiers of his time regarded with interest. If this 
conjecture b right, the mention of these angles would correspond to the 
insertion of definitions of which he makes no use, e.g. those of a rhombus and 
a rhomboid. 

Proclus has no hesitation in speaking of the " angle of a semicircle " and 
the "hom.like angle" as true angles. I'hus he says that "angles are contained 
by ■i. straight line and a circumference in two ways ; for they are either 
contained by a straight line and a cunve.Y circumference, like, that of the setni- 



40 BOOK III [ni. 1 6 

circle, or by a straight line and a concave circumference, like the mparodSift " 
(p. 127, II — 14). "There are mixed lines, as spirals, and angles, as the angle 
of a semicircle and the ntpaTOfiSij! " (p. 104, 16—18). The difficulty which 
the ancients felt arose from the very fact which Euclid embodies in this 
proposition. Since an angle can be divided by a line, it would seem to be a 
magnitude; "but if it is a magnitude, and all homogeneous magnitudes which 
are finite have a ratio to one another, then alt homogeneous angles, or rather 
all those on surfaces, will have a ratio to one another, so that the cornitular 
will also have a ratio to the rectilineal. But tilings which have a latio to one 
another can, if multiplied, exceed one another. Therefore the cornteular 
angle will also sometime exceed the rectilineal ; which ts impossible, for it is 
proved that the former is less than any rectilineal angle'' (Proclus, p. lai, 
24 — I2Z, 6). The nature of contact between straight lines and circles was 
also involved in the question, and that this was the subject of controversy 
before Euclid's time is clear from the title of a work attributed to Democritus 
(13. 420 — 400 B.C.) irtpi £((u^ap^i^9 yvu^oi/ot ^ irfpl ^axaxjOK icuicAtnf fmx trtpaiptjif 
On a differenu in a gnomon or on eontaei of a drcU and a sphert. There is, 
however, another reading of the first words of this title as given by Diogenes 
I^aertius (ex. 47), namely iripl Sta^p^i yfilfLiTt. On a difference of opinion, etc. 
May it not be that neither reading is correct, but that the words should be 
TTtpt Sto^p^i ytuci't/t t) jrtpt ^aiicTMx mJicXou not <r^<upt)!. On a difference in an 
angle or on contact with a circle and a sphere} There would, of course, 
hardly be any "angle" in conne>tion with the sphere; but I do not think that 
this constitutes any difficulty, because the sphere might easily be tacked on as 
a kindred subject to tiie circle. A curiously similar collocation of words 
appears in a passage of Proclus, though this may be an accident. He sa^s 
{p. 5*^* 4) "^^ ^i y*avi^v Bia^opOr^ ktyofJ4v nai aiffi^ftf auruit' ... and then, in 
the next hne but one, tt^ &i rav a^s rtay kukXiov ^ rmv cwfuiiv, " In what 
sense do we speak of differences of angles and of increases of than . . . and in 
what sense of the contaets (or meetings) of circles or of straight lines ? " 
I cannot help thinking that this subject of comicular angles would have had 
a fascination for Democritus as being akin to the question of infinitesimals, 
and very much of the same character as the other question which Plutarch 
{On Common Notions, xxxix. 3) says that he raised, namely that of the 
relation between the base of a cone and a section of it by a plane parallel to 
the base and apparenrty, to judge by the context, infinitely near to it : " if 
a cone were cut by a plane parallel to its base, what must we think of the 
surfaces of the sections, that they are equal or unequal? For, if they are 
unequal, they will make the cone irregular, as having many indentations like 
steps, and unevennesses ; but, if they are equal, the sections will be equal, 
and the cone will appear to have the property of the cylinder, as being made 
up of equal and not unequal circles, which is the height of absurdity." 

The contributions by Democritus to such investigations are further attested 
by a passage in the Method of Archimedes discovered by Heiberg In 1906 
{Archimedes, ed. Heiberg, Vol. ii. 191 3, p. 430; T. L. Heath, Tl'e Method 
of Archimedes, 1912, p. 13), which says that, though Eudoxus was the first to 
discover the scientific proof of the propositions (attributed to him) that the 
cone and the pyramid are one-third of the cylinder and prism respectively 
which have the same base and equal height, they were first stated, without 
proof, by Democritus. 

A full history of the later controversies about the cornicuiar " angle " 
cannot be given here ; more on the subject will be found in Camerer's 
Euclid (Excursus iv. on 11 1. 16) or in Cantor's Gtschichte der Maihematik. 



III. id} PROPOSITION i6 41 

Vol. 11. {see Contingenzwmkti in the index). But the following short note 
about the attitude of certain well-known mathematicians to the question will 
perhaps not be out of place, Johannes Campanus, who edited Euclid in 
the 13th century, inferred from [ti. 16 that there was a flaw in the principle 
that the iransitien from the less to the greater, or vice vers A, fakes place through 
all intermediate quantities and therefore through the egxial. If a diameter of a 
circle, he says, be moved about its extremity until it takes the position of the 
tangent to that circle, then, as lon^ as it cuts the circle, it makes an acute 
angle less than the " angle of a semicircle " ; but the moment it ceases to cut, 
it niakes a right angle greater than the same " angle of a semicircle." The 
rectilineal angle is never, during the transition, egual to the " angle of a semi- 
circle." There is therefore an apparent inconsistency with x. 1, and Campanus 
could only observe (as he does on that proposition), in explanation of the 
paradox, that " these are not angles in the same sense (univoce), for the 
curved and the straight are not things of the same kind without qualification 
(simpliciter)." The argument assumes, of course, that the right angle is 
greater than the "angle of a semicircle." 

Very similar is the statement of the paradox by Cardano (1501 — 1576), 
who observed that a quantity may eoutifiually increase without limit, and 
another diminish •without limit ; and yet the firsts however iTiereased, may be lest 
than the second, however diminished. The first quantity is of course the angle 
of contact, as he calls it, which may be " increased " indefinitely by drawing 
smaller and smaller circles touching the same straight line at the same point, 
but will always be less than any acute rectilineal angle however small. 

We next come to the French geometer, Peletier (Peletarius), who edited the 
Elements in r 557, and whose views on this subject seem to mark a great advance. 
Peletier's opinions and arguments are most easily accessible in the account of 
them given by Clavius (Christoph Klau[?], 1537 — 1612) in the 1607 edition of 
his Euclid. The violence of the controversy between the two will be understood 
from the fact that the arguments and counter-ai^uments (which sometimes run 
into other matters than the particular question at issue) cover, in that book, 
xt pages of small print. Peletier held that the " angle of contact " was not an 
angle at all, that the "contact of two circles," i.e. the "angle" between the 
circumferences of two circles touching one another internally or externally, is 
not a quantity, and that the " contact of a straight line with a circle " is not a 
quantity either; that angles contained by a diameter and a circumference 
whether inside or outside the circle are right angles and equal to rectilineal 
right angles, and that angles contained by a diameter and the circumference 
in all circles are equal The proof which Peletier gave of the latter pro- 
position in a letter to Cardano is sufficiently ingenious. If a greater and 
a less semicircle be placed with their diameters terminating at a common 
point and lying in a straight line, then (i) the angle ^the larger obviously 
cannot be less than the angle of the smaller. Neither (*) can the former be 
greater than the latter ; for, if it were, we could obtain another angle of a 
semicircle greater still by drawing a still larger semicircle, and so on, until we 
should ultimately have an angle of a. semicircle greater than a right angle ; which 
is imp)OSsible. Hence the angles ^semicircles must all be equal, and the dif- 
ferences between them nothing. Having satisfied himself that all angles of 
contact are JftfAangles, no^qu an titles, and therefore nothings, Peletier holds the 
difficulty about x. i to be at an end. He adds the interesting remark that 
the essence of an angle is in cutting, not contact, and that a tangent is not 
inclined to the circle at the point of contact but is, as it were, immersed in it at 
that point, just as much as if the circle did not diverge from it on either side. 



4» BOOK III [ill. 16 

The reply of Claviua need not detain us. He argues,' evidently appealing 
to the eye, that the angle of contact qan be divided by the arc of a circle 
greater than the given one, that the angles of two semicircles of different sizes 
cannot be equal, since they do not coincide if they are applied to one another, 
that there is nothing to prevent angia of coniact from being quantities, it being 
only necessary, in view of x. i, to admit that they are not of the same kind as 
rectilineal angles ; lastly that, if the angle of contact had been a nothing, 
Euclid would not have given himself so much trouble to prove that it is less 
than any acute angle. {The word is dtsudasset, which is certainly an 
exaggeration as applied to what is little more than an obiter dictum in in. id.) 

Vieta (1540 — 1603) ranged himself on the side of Peletter, maintaining 
that the angle of contact is no angle ; only he uses a new method of proof. 
The circle, he says, may be regarded as a plane figure with an infinite number 
of sides and angles ; but a straight line touching a straight line, however short 
it may be, will coincide with that straight line and wilt not make an angle. 
Never before, says Cantor (ii,, p. 540), had it been so plainly dccKired what 
exactly was to be understood by contact, 

Gahleo Galilei (1564 — 164*) seems to have held the same view as Vieta 
and to have supported it by a very similar argument derived from the com- 
parison of the circle and an inscribed polygon with an infinite number of 
sides. 

The last writer on the question who must be mentioned is John Wall is 
(1616— 1703}. He published in 1656 a paper entitled De angttlo contactus et 
semicireuli traetatus in which he also maintained that the so-cailed angle was 
not a true angle, and was not a quantity, Vincent Leotaud (1555—1672) 
took up the cudgels for Clavius in his Cyclomathia which appeared in 1663, 
This brought a reply from Wallis in a letter to Leotaud dated 17 February, 
1667, but not apparently published till it appeared in A defense of the treatise 
of the angle of contact which, with a separate title-page, and date 1G84, was 
included in the English edition of his Algebra dated 1685, The essence of 
Wallis' position may be put as follows. According to Euclid's definition, a 
plane angle is an inclination of two lines; therefore two lines forming an angle 
must incline to one another, and, if two lines meet without being inclined to 
one another at the point of meeting (which is the case when a circumference 
is touched by a straight line), the lines do not form an angle. The " angle of 
contact " is therefore no angle, because at the point of contact the straight line 
is not inclined to the circle but lies on it (1kA.lv<j9, or is coincident with it. 
Again, as a point is not a line but a heginning of a line, and a line is not a 
surface but a beginning oi a surface, so an angle is not the distance between 
two lines, but their initial tendency towards separation : Angulus {seu gradus 
divaricatianis) Distantia nen est sed Incef/thius distantiae. How far lines, which 
at their point of meeting do not fomt an angle, separate from one another as 
they pass on depends on the degree of curvature (gradus curvitatis), and it is 
the latter which has to be compared in the case of two lines so meeting. The 
arc of a smaller circle is more curved as having as much curvature in a lesser 
length, and is therefore curved in a greater degree. Thus what Clavius called 
angulus contactus becomes with Wallis gradus curvitatis, the use of which 
expression shows that curvature and curvature can be compared according to 
one and the same standard. A straight line has the least possible curvature ; 
but of the "angle" made by it with a curve which it touches we cannot say that 
it is greater or less than the " angle " which a second curve touching the same 
straight line at the same point makes with the first curve ; for in both ca.<«s 
there is no true angle at all (cf. Cantor m,, p. 24). 



lit. 16, 17] 



PROPOSITIONS 16, 17 



43 



The words usually given as a part of the corollary "and that a straight line 
touches a circle at one point only, since in fact the straight line meeting it in 
two points was proved to fall within it " are omitted by Hetberg as being an 
undoubted addition of Theon's. It was Simson who added the further remark 
that "it is evident that there can be but one straight line which touches the 
circle at the same point" 

Proposition 17. 

From a given point to draw a straight line touching a 
given circle. 

Let A be the given point, and BCD the given circle ; 
thus it is required to draw from the point A a straight line 
touching the circle BCD. 

For let the centre E of the circle 
be taken ; [m, i] 

let AE be joined, and with centre E 
and distance EA let the circle AEG 
be described ; 

from D Jet DF be drawn at right 
angles to EA, 
and let £F, AB h& joined ; 
I say that AB has been drawn from 
the point A touching the circle BCD. 

For, since E is the centre of the circles BCD, AFG, 
EA is equal to EF, and ED to EB ; 
therefore the two sides AE, EB are equal to the two sides 
FE, ED : 
and they contain a common angle, the angle at E ; 

therefore the base DF is equal to the base AB, 
.1 and the triangle DEE is equal to the triangle BE A, 

and the remaining angles to the remaining angles ; [i. 4] 
therefore the angle EDF is equal to the angle EBA. 

But the angle EDF is right ; 
therefore the angle EBA is also right. 

Now EB is a radius ; , 
and the straight line drawn at right angles to the diameter 
of a circle, from its extremity, touches the circle ; [in. 16, Por,] 
therefore AB touches the circle BCD. 

Therefore from the given point A the straight line AB 
has been drawn touching the circle BCD. 




m 



/ 



BOOK HI 



[ill. 17, 18 



The construction shows, of course, that two straight lines can be drawn 
from a givttn external point to touch a ^ven circle ; and it is equally obvious 
that these two straight lines are equal in length and equally inclined to the 
abaight line joining the exiemai point to the centre of the given circle. 
These facts are given by Heron {an-Nairlzl, p. 130). 

It is true that Euclid leaves out the case where the given point lies oit the 
circumference of the circle, doubtless because the construction is so directly 
indicated by iii. 16, For. as to be scarcely worth a separate statement. 

An easier solution is of course possible as soon as we know (ici. 31) that 
the angle in a semicircle is a right angle ; for we have only to describe a 
circle on AE as diameter, and this circle cuts the given circle in the two points 
of contact 



Proposition 18. 

// a straight tine touch a circle, and a straight line be 
jained from the centre to the point of cop fact, the straight line 
so joined wilt be perpendicular to the tangent. 

For let a straight line D£ touch the circle ABC at the 
point C let the centre F of the 
circle ABC be taken, and let FC 
be joined from F\o C; 
I say that FC is perpendicular to 
DE. 

For, if not, let FG be drawn 
from F perpendicular to DE. 

Then, since the angle FGC is 
right, 

the angle FCG is acute ; [i. 1 7] 
and the greater angle is subtended 
by the greater side ; [1. 19] 

therefore FC is greater than FG. 
But FC is equal to FB ; 
therefore FB is also greater than FG, 

the less than the greater: which is impossible. 
Therefore FG is not perpendicular to DE. 

Similarly we can prove that neither is any other straight 
line except FC ; 

therefore FC is perpendicular to DE. 
Therefore etc. 

Q. E, D. 




til, i8, 19] 



PROPOSITIONS 17—19 



4S 



3. the tangnit, 4 ifuwreiUr^. 

Just as 111. 3 contains two partial converses of the ForUm to lit. i, so 
the present proposition and the next give Jwo partial converses of the 
corollary to iii, 16, We may show their relation thus: suppose three things, 
( r) a tangent at a jwint of a circle, (2) a straight line drawn from the centre to 
the point of contact, (t) right angles made at the point of contact [with (i) or 
{1) as the case may bej. Then the corollary to in. 16 asserts that (t) and (3) 
together give (i), iii. 18 that (t) and (i) give (3), and iii. rg that (t) and (3) 
give (1), i.e. that the straight line drawn from the point of contact at right 
angles to the tangent passes through the centre. 




Proposition 19. 

If a straight line iottch a circle, and from the point of 
contact a straight line be drawn at right angles to the tangent, 
the centre of the circle will lie on the straight line so dratvn. 

For let a straight line DE touch the circle ABC at the 
point C, and from C let CA be 
drawn at right angles to DE ; 
I say that the centre of the circle 
is on A C. 

For suppose it is not, but, if 
possible, let F be the centre, 
and let CF be joined. 

Since a straight line DE touches 
the circle ABC, 

and FC has been joined from the 
centre to the point of contact, 

FC is perpendicular to DE ; [111, 18] 

therefore the angle FCE is right 

But the angle ACE is also right ; 

therefore the angle FCE is equal to the angle ACE, 
the less to the greater : which is impossible. < 

Therefore F is not the centre of the circle ABC. 

Similarly we can prove that neither is any other point 
except a point on AC. 

Therefore etc. 

Q. E. D. 

We may abo regard iii, 19 as a partial converse of in. 18. Thus suppose 
(t) a straight line through the centre, (s) a straight line through the point of 
contact, and suppose (3) to mean perpendicular to the tangent ; then iii. tS 
asserts that (i) and (2) combined produce (3}, and 111. 19 that (1} and (3) 



4& / »• BOOK Iir ^.1'' [111.19,30 

produce (i); while again we may enundate a second partial converse of iii, 18, 
corresponding to the statement that (i) and (3) produce (2), to the effect that 
a straight line drawn through the centre perpendicular to the tangent passes 
through the point of contact. 

We may add at this poin^ or even after the Porism to ill, 16, the theorem 
that ttuo circles which touch om another internally or externally have a common 
tangent at their point of cotttaei. For the line joining their centres, produced 
if necessary, passes through their point of contact, and a straight line diawn 
through that point at right angles to the line of centres is a tangent to both 
circles. 



Proposition 20. 

In a circle the angle at ihe centre is double of the angle 
at the circumference, when the angles have the same circum- 
ference as base. 

Let ABC be a circle, let the angle BEC be an angle 
sat its centre, and the angle BAC an 
angle at the circumference, and let 
them have the same circumference BC 
as base ; 

I say that the angle BEC is double of 
10 the angle BAC. 

For let AE be joined and drawn 
through to F. 

Then, since EA is equal to EB, 
the angle EAB is also equal to the 
IS angle EBA ; [1. 5] 

therefore the angles EAB, EBA are double of the angle 
EAB. 

But the angle BEF is equal to the angles EAB, EBA ; 

['■ 3-1 
therefore the angle BEF is also double of the angle 
taEAB. 
' For the same reason 

the angle FEC is also double of the angle EAC. 
Therefore the whole angle BEC is double of the whole 
angle BAC. 
as Again let another straight line be inflected, and let there 
be another angle BDC\ let DE be joined and produced 
to G, 




111. »o] PROPOSITIONS 19, 10 47 

Similarly then we can prove that the angle GEC is 
double of the angle EDC, 
» of which the angle GEB is double of the angle EDB ; 

therefore the angle BEC which remains is double of the 
angle BDC. 

Therefore etc. Q. e. d. 

15. let another straight line be inflected, atM.tin ik ir(Ui> (without (Wcia). The 
verb jfXdw (to brtak off) was the regular technical term for drawln^r from a point a (broken) 
straight line which hfst tneeis another straight line or curve and is then htnt lnuk ham it 
to anmher point, or (in other words) for drawing .straight lines from two points meeting at a 
point on a curve or another straight line. Ki>t\Aff9v^t Li one of the geometrical terms ^he 
definition of which must according to Aristotle be assumed [AtuiL Rat, t. \o, 76 b 9). 

The early editors, Tartaglia, Commandinus, Peletarius, Clavius and others, 
gave the extension of (his proposition to the case where the segment is less 
than a semicircle, and where accordingly the " angle " corresponding to 
Euclid's " atigle at the centre " is greater than two right angles. The 
convenience of the extension is obvious, and the proof of it is the same as the 
first part of Euclid's proof. By means of the extension in. 2 1 is demonstrated 
without making two cases; Jti. zz will follow immediately from the fact that 
the sum of the " angles at the centre " for two segments making up a whole 
circle is equal to four right angles; also 111. 31 follows immediately from the 
extended proposition. 

But all the editors referred Xq were forestalled in this matter by Heron, as 
we now learn from the commentary of an-Naitizi (ed. Curtxe, p. 131 sqq.). 
Heron gives the extension of Euclid's pro[>osition which, he says, it had been 
left for him to make, but which is necessary in order that the caviller may not 
be able to say that the next proposition (about the equality of the angles 
in any segment) is not established generally, i.e. in the case of a segment less 
than a semicircle as well as in the case of a segment greater than a semicircle, 
inasmuch as lit. no, as given 'ay Euclid, only enables us to prove it in the 
latter case. Heron's enunciation is imt>ortaiit as showing how he describes 
what we should now call an " angle " greater than two right angles. (The 
language of Gherard's translation is, in other respects, a little obscure ; but 
tht: meaning is made clear by what follows.) 

"The angle," Heron says, "which is at the centre of any circle is double 
of the angle which is at the circumference of it vrhen one arc is tk( base of bolk 
angles; and the remaining angles whieh are at I he centre, and fill up the four 
right angles, are double of the angle at the circumference of the ate which is 
subtended by the [original] angle which is at the ceiitre," 

Thus the " angle greater than two right angles " is for Heron the sum of 
certain "angles" in the Euclidean sense of angles less than two right angles. 
The particular method of splitting up which Heron adopts will be seen from 
his proof, which is in substance as follows. 

r Let CDB be an angle at the centre, CAB that at the circumference. 
' Produce SD, CDto F,G; 

take any point jE on BC^ and join BE, EC, ED. 

Then any angle in the segment BAC is half of the angle SDC; and 
tht turn of the angles BDG, GDF, FDC is double of any angle in the 
segment BEC. 




48 -I BOOK in [til. 10 

I'ttof. Since CZ7 is equal, to £Z7, I't./ ■• ,tuiM v i - 

the angles DCE, DEC art equal. 

Therefore the exterior angle GDE is equal to 
twice the angle DEC. 

Similarly the exterior angle EDE is equal to 
twice the angle DEB. 

By addition, the angles GDE, EDE are double 
of the angle BEC. 

But 
the angle BDC is equal to the angle EDG, 

therefore the sum of the anglts BDG, GDF, FDC 
is doubU of the angle BEC. 

And Euclid has proved the first part of the 
proposition, namely that the angle BDC is double 
of the angle BAC. 

Now, says Heron, BAC is any angle in the segment BAC, and therefore 
any angle in the segment BAC is half of the angle BDC. 

Therefore all the angles in the segment BAC are equal. 

Again, BEC is any angle in the segment BEC and is equal to half the 
sum of the angles BDG, GDF, FDC. 

Therefore all the angles in (he segment BEC are equal 

Hence in. 2 1 is proved generally. 

Lastly, says Heron, 
since the sum of the angles BUG, GDF, FDC is double of the angle BEC, 
and the angle BDC is double of the angle BAC, 

therefore, by addition, the sum of four right angles is double of tKe sum of 
the angles .^.^C, BEC. 

Hence the angles BAC, BEC are together equal to two right angles, and 
III. 12 is proved. 

The above notes of Heron show conclusively, if proof were wanted, that 
Euclid had no idea of in. zo applying in terms (either as a matter of 
enunciation or proof) to the case where the angle at the circumference, or the 
angle in the segmenl^ is oituse. He would not have recognised the " angle " 
greater than two right angles or the so-called "straight angle" as being an 
angle at all. This is indeed clear from his definition of an angle as the 
ittciinatien ic.r.i,, and from the language used by other later Greek mathe- 
maticians where there would be an opportunity for introducing the extension. 
Thus Proclus' notion of a "four-sided triangle" (cf the note above on the 
definition of a triangle) shows that he did not count a re-entrant angle as an 
angle, and Zenodorus' application to the same figure of the word "hollow- 
angled " shows that in that case it was the exterior angle only which he would 
have called an angle. Further it would have been inconvenient to have 
introduced at the beginning of the Elements an "angle" equal to or greater 
than two right angles, because other definitions, e;g. that of a right angle, 
would have needed a qualification. If an "angle" might be equal to two 
right angles, one straight line in a straight line with another would have 
satisfied Euclid's definition of a right angle. This is noticed by Dodgson 
(p. 160), but it is pmctically brought out Dy Proclus on i, 13. "For he did 
not merely say that ' any straight line standing on a straight line either makes 
two right angles or angles equal to two right angles ' but ' if it make angles.' 



III. 20, »i) PROPOSITIONS 30, Ji 49 

If it stand on the straight line at it$ extremity and make one angle, is it 
possible for this to be equal to two right angles ? It is of course impossible ; 
jbr every rectUineai angle is iess than two right angles, as every solid angle is 
less than four right angles (p. 291, 13 — 20)." [It is (rue that it has been 
generally held that the meaning of " angle " is tacitly extended in vi. 33, but 
there is no real ground for this view. See the note on the proposition.! 

It will be observed that, following his usual habit, Euclid omits the 
demonstration of the case which some editors, e.g. Clavius, have thought it 
necessary to give separately, the case namely where one of the lines forming 
the angle in the s^ment passes through the centre. Euclid's proof gives so 
obviously the means of proving this that it is properly left out. 

Tod hunter observes, what Clavius had also remarked, that there are two 
assumptions in the proof of 111. zo, namely that, if A is double of B and C 
double of D, then the sum, or difference, of A and C is equal to double the 
sum, or difference, of B and D respectively, the assumptions being particular 
cases of v. i and v. 5. But of course it is easy to satisfy ourselves of the 
correctness of the assumption without any recourse to Book v. 

' ' ' Proposition 21. 

In a circle the angles in ike same segment are equal to one 
another. 

Let A BCD be a circle, and let the angles BAD, BED 
be angles in the same segment BAED ; 
I say that the angles BAD, BED are 
equal to one another. 

For let the centre of the circle 
ABCD be taken, and let it be .f ; let 
BE, ED be joined. 

Now, since the angle BED is at 
the centre, 

and the angle BAD at the circum- 
ference, 

and they have the same circumference BCD as base, 
therefore the angle BED is double of the angle BAD. [m. 20] 

For the same reason 

the angle BFD is also double of the angle BED ; 
therefore the angle BAD is equal to the angle BED. 
/ Therefore etc, 

, Q. E. D. 

Under the restriction that the " angle at the centre " used in iii, *o must 
be less than two right angles, Euclid's proof of this proposition only applies 
to the case of a segment greater than a semicircle, and the case of a segment 
equal to or less than a semicircle has to be considered separately. The 
simplest proof, of many, seems to be that of Simson. 




s* 



BOOK in 



[hi. 21 




" But, if the segment BARD be not greater than a semicircle, let SAD, 
BED be angles in it: these also are equal to one 
another. 

Draw AJ^lQ the centre, and produce it to C, and 
join CE. 

Therefore the segment BADC is greater than a 
semicircle, and the angles in it SAC, EEC are equal, 
by the first case. 

For the same reason, because CBED is greater 
than a semicircle, 

the angl^ CAD, CED are equal. 

Therefore the whole angle BAD is equal to the whole angle BED." 

We can prove, by means of redudio ad absurdiim, the important converse 
of this proposition, namely that, if there be any hoo Iriangies on the same base 
and on the same side of it, and with equal vertical angles, the circle passing 
through the extremities of the base and the vertex of one triangle will pass 
through the vertex of the other triangle also. That a circle can be thus 
described about a triangle is clear from Euclid's construction in itt, 9, which 
shows how to draw a circle passing through any three points, though it is 
in tv. 5 only that we have the problem stated. Now, 
suppose a circle BAC drawn through the angular 
points of a triangle BAC, and let BDC be another 
triangle with the same base BC and on the same side 
of it, and having its vertical angle D equal to the 
angle A. Then shal! the circle pass through D. 

For, if it does not, it must pass through some point 
E on BD or on BD produced. If then EC be 
joined, the angle BEC is equal to the angle BAC, 
by in. %\, and therefore equal to the angle BDC. 
Therefore an exterior angle of a triangle is equal to 
the interior and opposite angle; which is impossible, by 1. 16. 

Therefore D lies on the circle BA C. 

Similarly for any other triangle on the base BC and with vertical angle 
equal to A. Thus, if any number of triangles be constructed on the same base 
and on the same side of it, with equal vertical angles, the vertices will all lie on 
the circumfererue of a segmetit of a circle. 

A useful theorem derivable from ill. 21 is given by Serenus (ZV sectUme 
coni. Props. 5?, 53). 

If ADB be any segment of a circle, and C l>e such a point on the 
circumference that AC \i equal to CB, and if 
there be described with C as centre and radius 
CA or CB the circle AI/B, then, ADB being 
any other angle in the segment ACB, and BD 
being produced to meet the outer segment in 
E, the sum of AD, DB is equal to BE. 

If BC be produced to meet the outer 
segment in F, and FA be joined, 

CA, CB, CEaie by hypothesis equal. 

Therefore the angle EAC is equal to the 
angle AEC. 

Also, by Itt. It, the angles ACB, ADB are equal ; 





III. ai, 12] PROPOSITIONS 11, a* 51 

therefore their supplements, the angles jiC/^, AJ}£, are equal 

Further, by m. ai, the angles AEB, AFB are equal. 

Hence in the triangles ACF, ADE two angles are respectively equal ; 

therefore the third angles EAD, FAC are equal. 

But the angle FAC is equal to the angle AFC, and therefore equal to the 
angle AED. 

Therefore the angles AED, EAD are equal, or the triangle DEA is 
isosceles, 

and AD is equal to DE. 

Adding BD to both, we see that 

BE is equal to the sum of AD and DB. 

Now, £F being a diameter of the circle of which the outer segment is 
a part, 

BF is greater than BE ; 

therefore AC, CB are together greater than AD, DB. 

And, generally, of ail trianglts oh tht same bas€ and on the same side of it 
whkh hat>e tqttai vertical angles, the isosales triangle is that whieh has the 
greatest perimeter, and of the others that has the lesser perimeter which is 
further from being isosceles. 

The theorem of Serenus gives us the means of solving the following 
problem given in.Todhunter's Euclid, p 314. 

To find a point in the eirmmferenee of a given segment of a circle such that 
the straight Una which Jain the point to the extremities of the straight line on 
which tlu segment stands may ^ together equal to a given straight line (the 
length of which is of course subject to limits). 

Let A CB in the above figure be the given segment. Find, by bisecting 
AB at right angles, a point C on it such that ^ C is equal to CB. 

Then with centre C and radius CA or CB describe the segment of a 
ctccle AHB on the satne side of AB. 

Lastly, with ^4 or i' as centre and radius equal to the given straight line 
describe a circle. This circle will, if the given straight line be greater than 
AB and less than twice AC, meet the outer segment in two points, and if we 
join those points to the centre of the circle last drawn (whether A or B\ the 
joining straight lines will cut the inner segment in points satisfying the given 
condition. If the given straight line be eguai to twice AC, C is of course 
the required point. If the given straight line be greater than twice .^JC, there 
is no possible solution. 

' ' ■■ ' ' Proposition 22, 

The opposite angles of qiutdrilaterah in circles are equal 
to two right angles. 

Let ABCD be a circle, and let ABCD be a quadrilateral 
in it; 
1 say that the opposite ang^t.3 are equal to two right angles. 

Let AC, BD be joined. 

Then, since in any triangle the three angles are equal to 
two right angles, [1. 31] 




S» BOOK III [ill. *J, a3 

the three angles CAB, ABC, BCA of the triangle ABC 
are equal to two right angles. 

But the angle CAB is equal to the 
angle BDC, for they are in the same 
segment BADC; [m. ii] 

and the angle ACS is equal to the angle 
ADB, for they are in the same segment 
ADCB', 

therefore the whole angle ADC is equal 
to the angles BAC, ACB. 

Let the angle ABC be added to each ; i ^ 

therefore the angles ABC, BAC, ACB are equal to the 
angles ABC, ADC. 

But the angles ABC, BAC, ACB are equal to two right 
angles ; 

therefore the angles ABC, ADC are also equal to two right 
angles. 

Similarly we can prove that the angles BAD, DCB are 
also equal to two right angles. 

Therefore etc, 

Q. E. D. 

As Todhunter remarks, the converse of this proposition is true and very 
important : if hvo opposite anglts of a quadrilaitral bt togeShtr equal to two 
right angin, a dnk may de (ircumsmbed about the quadrilateral. We can, by 
the method of in. 9, or by iv. 5, circumscribe 3 circle about the triangle 
ABC; and we can then prove, by reductio ad nbsurdum, that the circle 
passes through the fourth angular fioint D. 

• ,.15 . 

Proposition 23. 

On the same straight line there cannot be constructed two 
similar and unequal segments of circles on the same side. 

For, if possible, on the same straight line AB let two 
similar and unequal segments of circles 
ACB, ADB be constructed on the same 
side ; ^-^ii^^^^^ ° 

let A CD be drawn through, and let CB, 
DB be joined. 

Then, since the segment ACB is 
similar to the segment ADB, 




III. »3, 24] PROPOSITIONS aa— 14 53 

and similar segments of circles are those which admit equal 

angles, [m. Def. u] 

the angle ACB is equal to the angle ADB, the exterior 
to the interior : which is impossible. , [i. 16] 

Therefore etc. 

I. cannot be conatnicled, ti nwro^rrcu, the stmt phnae is in [. 7. 

Clavius and the other early editors point out that, while the words "on 
the same side " in the enunciation are necessary for Euclid's proof, it is 
equally true that neither can there be two similar and unequal segments on 
apposite sides of the same straight line ; this is at once made clear by causing 
one of the segments to revolve round the base till it is on the same side with 
the other. 

Simson observes with reason that, while Euclid in the following proposition, 
III. 24, thinks It necessary to dispose of the hypothesis that, if two similar 
segments on equal bases are applied to one another with the bases coincident, 
the segments cannot cut in any other jwint than the extremities of the base 
(since otherwise two circles would cut one another in more points than two), 
this remark is an equally necessary preliminary to iii. 23, in order that we 
may be justified in drawing the segments as being one inside the other- 
Sim son accordingly begins his proof of in. 23 thus : 

"Then, because the circle ACB cuts the circle ADB in the two points 
A, B, they cannot cut one another in any other point : 

One of the segments must therefore fall within the other. 

Let ACS fall within ADB and draw the straight line ACI), etc." 

Simson has also substituted "not coinciding with one another" for 
"unequal" in Euclid's enunciation. 

Then in i[i. 24 Simson leaves out the words referring to the hypothesis 
that the segment AEB when applied to the other CFD may be " otherwise 
placed as CGD" \ in fact, after stating that AS must coincide with CD, he 
merely adds words quoting the result of ni. aj : "Therefore, the straight line 
j4.^ coinciding with CD, the segment AEB must coincide with the segment 
CFD, and is therefore equal to it." 



Proposition 24. 

Similar segments of circles on equal straight lines are equal 
to one another. 

For let AEB, CFD be similar segments of circles on 
eqtial straight lines AB, CD ; 
s I say that the segment AEB is equal to the segment CFD. 

For, if the segment AEB be applied to CFD, and if the 
point A be placed on C and the straight line AB on CZ?, 



54 BOOK in "" [hi. 34, *s 

the point B will also coincide with the point D, because 
AB\& equal to CD ; 

10 and, ^^ coinciding with CD, ' • ; .• . ■ 1 

. , ' •. !;'r(j 

the segment AEB will also coincide with CFD. ' r 





For, if the straight line AB coincide with CD but the 
segment AEB do not coincide with CFD, 

it will either fall within it, or outside it ; • i- 

IS or it will fall awry, as CGD, and a circle cuts a circle at more 
points than two : which is impossible. [m. 10] 

Therefore, if the straight line AB be applied to CD, the 
segment AEB will not fail to coincide with CFD also ; 

therefore it will coincide with it and will be equal to it. 

20 Therefore etc, 

' ' • • Q. E. D. 

rj . fftU awiy, TopoWd^fc, the same v/ord a& used in tfae Uke case in [. S. The word 
impHes that the applied figure will partly fall short of, and partly overlap, the Aguie to 
which it is applied^ 

Compare the note on the last proposition. I have put a semicolon instead 
of the comma which the Greek text has after "outside it," in order the better 
to indicate that the inference "and a circle cuts a. circle in more points than 
two " only refers to the third hypothesis that the applied segment is "otherwise 
placed (ifapaAXftfft) as CGD." The first two hypotheses are disposed of by 
a tacit reference to the preceding proposition in. 23, 



Proposition 25. 

Given a segment 0/ a circle, to describe the complete circle 
ofiuhich it is a segment. 

Let ABC be the given segment of a circle ; 

thus it is required to describe the complete circle belonging 
to the segment ABC, that is, of which it is a segment. 

For let AC he. bisected at D, let DB be drawn from the 
point D at right angles to AC, and let AB, be joined ; 




uj. as] PROPOSITIONS 24, 25 55 

the angle ABD is then greater than, equal to, or less 
than the angle BAD. 

First let it be greater ; ' ' ' "^ 

and on the straight line BA, and at the point A on it, let 
the angle BAE be constructed equal to 
the angle ABD; let DB be drawn through 
to E, and let EC be joined. 

Then, since the angle ABE is equal to 
the angle BAE, 

the straight line EB is also equal to 
EA. [1. 6] 

And, since AD is equal to DC, 
and DE is common, ., . -< 1 

the two sides AD, DE are equal to the two sides CD, DE 
respectively ; 

and the angle ADE is equal to the angle CDE, for each is 

right ; . ,, 

therefore the base AE is equal to the base C£. • 

But AE was proved equal to BE ; 

therefore BE is also equal to CE ; 

therefore the three straight lines AE, EB, EC are equal to 
one another. 

Therefore the circle drawn with centre E and distance 
one of the straight lines AE, EB, EC will also pass through 
the remaining points and will have been completed, [ni. 9] 

Therefore, given a segment of a circle, the complete circle 
has been described. 

And it is manifest that the segment ABC is less than a 
semicircle, because the centre E happens to be outside it. 

Similarly, even if. the angle ABD be equal to the angle 
BAD, 

AD being equal to each of the two BD, DC, 

the three straight lines DA, DB, DC will 
be equal to one another, 

D will be the centre of the completed circle, 

and ABC will clearly be a semicircle. 





56 BOOK III ' [ill. 35, 3fi 

But, if the angle ABD be less than the angle BAD, 
and if we construct, on the straight line BA 
and at the point A on it, an angle equal to 
the angle ABD, the centre will fall on DB 
within the segment ABC, and the segment 
ABC will clearly be greater than a semi- 
circle. 

Therefore, given a segment of a circle, 
the complete circle has been described. 

Q. E. F. 

T. to deaciibe the complete circle^ itpatravaypd^pat tov kAxXov, Utenlly "Lo describe 
the circle tm fo it. * 

It will be remembered that Sim son takes first the case in which the angles 
ABD, BAD are equal to one another, and then takes the other two cases 
together, telling us to "produce BD, if necessary." This is a little shorter 
than Euclid's procedure, though Euclid does not repeat the proof of the first 
case in giving the third, but only refers to it as equally applicable. 

Campanus, Petetarius and others give the solution of this problem in 
which we take two chords not parallel and bisect each at rj^ht angles by 
straight lines, which must meet in the centre, since each contains the centre 
and they only intersect in one point. Clavius, Billingsley, Barrow and others 
give the rather simpler solution in which the two chords have one extremity 
common (cf. Euclid's proofs of lit. g, ro). This method De Morgan favours, 
and (as noted on in. i above) would make iii, i, this proposition, and 
IV, 5 all coroilaries of the theorem that " the line which bisects a chord 
perpendicularly must contain the centre," Mr H. M. Taylor practically 
adopts this order and method, though he finds the centre of a circle by 
means of any two non -parallel chords ; but he finds the ctntre of the circle of 
whkh a given art is a part (his proposition corresponding to in. 15) by 
bisecting at right angles first the base and then the chord joining one extremity 
of the base to the point in which the line bisecting the base at right angles 
meets the circumference of the segment. Under De Morgan's alternative the 
relation between Euclid in. i and the Porism to it would be reversed, and 
Euclid's notion of a Porism or corollary would have (o be considerably 
extended. 

If the problem is solved ^fter the manner of iv. 5, it is still desirable to 
state, as Euclid does, after proving AE, EB, EC to be all equal, that "the 
circle drawn with centre E and distance one of the straight lines AE, EB, 
EC will also pass through the remaining points of the segment" [ni. 9], in 
order to show that part of the circle described actually coincides with the 
given segment. This is not so clear if the centre is determined as the 
intersection of the straight lines bisecting at right angles chords which join 
pairs of four different points. 

Proposition 26. 

fn equal arcles equal angles stand on equal arcumferenees, 
wheUier they stand at tlie centres or at the circumferences. 



III. a6] PROPOSITIONS 25, 26 57 

Let ABC, DEF be equal circles, and in them let there 
be equal angles, namely at the centres the angles BGCt 
EHF, and at the circumferences the angles BAC, EDF\ 
I say that the circumference BKC is equal to the circum- 
ference ELF. 





■' 1,1 ■\ 

For let i?C, ^^ be joined. • . 

Now, since the circles ABC, DEFz.re equal, 

the radii are equal. 
Thus the two straight lines BG, GC are equal to the 
two straight lines EH, HF; 

and the angle at G is equal to the angle at H; 
therefore the base BC is equal to the base EF. [1. 4] 
And, since the angle at A is equal to the angle at D, 
the segment BAC is similar to the segment EDF; 

[hi. Def. 11] 
and they are upon equal straight lines. 

But similar segments of circles on equal straight lines are 
equal to one another ; [in. 34] 

therefore the segment BAC is equal to EDF. 
But the whole circle ABC is also equal to the whole circle 
DEF: 

therefore the circumference BKC which remains is equal to 
the circumference j£'Z/*'. ,.-.,... 

Therefore etc. Q, e. d. 

As in HI. 21, if Euclid's ptoof is to cover all cases, it requires us to take 
cognisance of " angles at the centre " which are equal w or greater than two 
■right angles. Otherwise we must deal separately with the cases where the 
angle at the circumference is equal to or greater than a right angle. The 
case of an ebtust angle at the circumference can of course be reduced by 
means of ni. iz to the case of an acute angle at the circumference; and, in 
case the angle at the circumference is right, it is readily proved, by drawing 
the radii to the vertex of the angle and to the other extremities of the lines 
containing it, that the latter two radii are in a straight line, whence they make 
equal bases in the two circles as in Euclid's proof. 



SS BOOK III [lit. 36.17 

Lordner has another way of dealing with the right angle or obtuse angte 
at the circumference. In either case, he says, "bisect them, and the halves 
of them are equal, and it can be proved, as above, that the arcs upon which 
these halves stand are equal, whence it follows that the arcs on which the 
given angles stand are equal." 



Proposition 27, 

/« egua^ circles angles standing on equal circumferences 
are equal ta one another, whether they stand at the centres or 
at the circumferences. 

For in equal circles ABC, DBF, on equal circumferences 
BC, EF, let the angles BGC, EHF stand at the centres G, 
H, and the angles BAC, EDF 3X the circumferences ; 

I say that the angle BGC is equal to the angle EHF, 

and the angle BAC is equal to the angle EDF. 




For, if the angle BGC is unequal to the angle EHF, 

one of thetn is greater. 

Let the angle BGC be greater : and on the straight line BG, 

and at the point G on it, let the angle BGK be constructed 

equal to the angle EHF. [1. as) 

Now equal angles stand on equal circumferences, when 
they are at the centres ; [m. a6] 

therefore the circumference BK is equal to the circum- 
ferencer EF. 

But EF is equal to BC ; 

therefore BK is also equal to BC, the less to the 
greater : which is impossible. 

Therefore the angle BGC is not unequal to the angle 
EHF; 

therefore it is equal to it 



HI. a7, »8] PROPOSITIONS 16—28 $9 

And the angle at A is half of the angle BGC, 
and the angle at D half of the angle EHF\ [m. ao] 

therefore the angle at A is also equal to the angle at D. 
Therefore etc. 

Q. E. D. 

This proposition is the converse of the preceding one, and the remarks 
about the method of treating the different cases apply here also. 



Proposition 28. 

In egtial circles equal straight lines cut off equal circum- 
ferences, the greater equal to the greater and Ike less to tlie 
less, .■;, 

■ - ..'1. ' <T ' • 

Let ABC, DEF be equal circles, and in the circles let 
AB, DE be equal straight lines cutting off ACS, DEE as 
greater circumferences and AGB, DHE as lesser; 
I say that the greater circumference ACB is equal to the 
greater circumference DFE, and the less circumference ^G^jff 
to DHE. 





For let the centres K, L of the circles be taken, and let 
^ A", ^^, Z?Z., Z.^ be joined. . -, v, -i ^./ j- 

Now, since the circles are equal, . . 

the radii are also equal ; 
therefore the tjvo sides AK, KB are equal to the two 
sides DL, LE ; 
and the base AB is equal to the base DE ; 

therefore the angle A KB is equal to the angle DLE. 

[I. 8] 
But equal angles stand on equal circumferences, when 
they are at the centres ; [m. 26] 

therefore the circumference AGB is equal to DHE. 



6ai « BOOK. Ill ";^•^ [HI. a8, J9 

And the whole circle ABC is also equal to the whole 
circle DEF\ 

therefore the circumference ACB which remains is also equal 
to the circumference DFE which remains. 

Therefore etc. 

Q. E. D, 

Euclid's proof does not in terms cover the particular case in which the 
chord in one circle passes through its centre ; but indeed this was scarcely 
worth giving, as the proof can easily be supplied. Since the chord in one 
circle passes through its centre, the chord in the second circle must also be a 
diameter of that circle, for equal circles are those which have equal diameters, 
and all other chords in any circle are less than its diameter [in. 15]; hence 
the segments cut off in each circle are semicircles, and these must be equal 
because the circles are equal. 

Proposition 29. 

In equal circles equal circumferences are subtended by equal 
straight lines. 

Let ABC, DBF be equal circles, and in them let equal 
circumferences BGC, EHF be cut off; and let the straight 
lines BC, EF be joined ; 
I say that BC is equal to EF. 





For let the centres of the circles be taken, and let them 
be a: Z ; let BK, KC, EL, Z/^ be joined. 

Now, since the circumference BGC is equal to the 
circumference EHF, 

the angle BKC is also equal to the angle .£'Z/^ [m. 17] 
And, since the circles ABC, DEF are equal, 

the radii are also equal ; 
therefore the two sides BK^ KC are equal to the two sides 
EL, LF; and they contain equal angles ; 

therefore the base BC is equal to the base EF. [i. 4] 
Therefore etc. 




iti. 29—31] PROPOSITIONS *8— 31 61 

The particular case of this converse of ill. 28 in which the given arcs are 
ares of semicircles is even easier than the corresponding case of in, 18 itself. 

The propositions in, z6 — 29 are of couise equaliy true if the same circle 
is taken instead of iwo equal circles. 



Proposition 30. 
To bisect a given circumference. 
Let ADB be the given circumference ; 
thus it is required to bisect the circumference ADB. 

Let AS h^ joined and bisected at 
C ; from the point C let CD be drawn o 

at right angles to the straight line AB, 
and let AJJ, DB be joined. 

Then, since ACb^ equal to CB, 
and CD is common, 

the two sides A C, CD are equal to the two sides BC, CD ; 

and the angle ACD is equal to the angle BCD, for each is 
right ; 

therefore the base AD is equal to the base DB. [1. 4] 

But equal straight lines cut off equal circumferences, the 
greater equal to the greater, and the less to the less ; [in- «8] 

and each of the circumferences AD, DB is less than a 
semicircle ; 

therefore the circumference AD is equal to the circum- 
ference DB, 

Therefore the given circumference has been bisected at 
the point D. 

Q. E. F. 

; ' Proposition 31. - , ,, , •' 

In a circle the angle in the semicircle is right, that in a 
greater segment less than a right angle, and that in a less 
segment greater than a right angle ; and further the angle of 
the greater segment is greater than a right angle, and the angle 
of the less segment less than a right angle. 




«» BOOK m [ill. 31 

- Let ABCD be a circle, let BC be its diameter, and E its 
centre, and let BA, AC, AD, DC 
be joined ; 

I say that the angle BAC in the 
semicircle BAC is right, 
the angle ABC in the segment -^^C 
greater than the semicircle is less 
than a right angle, 
and the angle ADC in the segment 
ADC less than the semicircle is 
greater than a right angle. 

Let AE be joined, and let BA ,„ ... , ; 

be carried through to ^. , ., , . i 

Then, since BE is equal to EA^ 

the angle A BE is also equal to the angle BAE, [1. s] 
; Again, since CE is equal to EA, ■..-^^i^ ■ /r* 

the angle ACE is also equal to the angle CAE. [1. s] 

Therefore the whole angle BAC is equal to the two angles 
ABC, ACB. 

But the angle EAC exterior to the triangle ABC is also 
equal to the two angles ABC, ACB ; [i. 31] 

therefore the angle BAC is also equal to the angle EAC; 

therefore each is right ; [1. Def. 10] 

therefore the angle BAC in the semicircle BAC is right. 

Next, since in the triangle ABC the two angles ABC, 
BAC are less than two right angles, [i. 17] 

and the angle BAC is a right angle, 

the angle ABC is less than a right angle ; 
and it is the angle in the segment ABC greater than the 
semicircle. 

Next, since ABCD is a quadrilateral in a circle, 
and the opposite angles of quadrilaterals in circles are equal 
to two right angles, [iil »] 

while the angle ABC is less than a right angle, 
therefore the angle ADC which remains is greater than a 
right angle ; 

and it is the angle in the segment ADC less than the semi- 
circle. 



HI. 31] PROPOSITION 31 63 

I say further that the angle of the greater segment, namely 
that contained by the circumference ABC and the straight 
line AC, is greater than a right angle ; 

and the angle of the less segment, namely that contained by 
the circumference ADC and the straight line AC, is less than 
a right angle. 

This is at once manifest. 
For, since the angle contained by the straight lines BA, AC 
is right, 

the angle contained by the circumference ABC and the 
straight line AC is greater than a right angle. 

Again, since the angle contained by the straight lines 
AC, AFis right, 

the angle contained by the straight line CA and the 
circumference ADC is less than a right angle. 

Therefore etc. q. e. d. 

As already stated, this proposition is immediately deducible from in. 20 if 
that theorem Is extended so as to include the case where the segment is equal 
to or less than a semicircle, and where consequently the " angle at the centre" 
is equal to two right angles or greater than two right angles respectively. 

There are indications in Aristotle that the proof of the first part of the 
theorem in use before Euclid's time proceeded on different lines. Two 
passages of Aristotle refer to the proposition that the angle in a semicircle 
IS a right angle. The first passage is Anal. Fast 11. 11, 94 a 38: "Why is 
the angle in a semicircle a right arvgle? Or what makes it a right angle? 
(tivo! ovrtK op$ij;) Suppose ^4 to be a right angle, B half of two right 
angles, C the angle in a semicircle. Then B is the cause of j4, the right 
angle, being an attribute of C, the angle in the semicircle. For £ is equal to 
/*, and CtoB; for C is half of two right angles. Therefore it is in virtue of 
£ being half of two right angles that A is an attribute of C ; and the latter 
means the fact that the angle in a semicircle is right." Now this passage 
by itself would be consistent with a proof like Euclid's or the alterrmtive 
interpolated proof next to be mentioned. But the second passage throws a 
different light on the subject. This is Metaph, 1051 a 26 ; "Why is the angle 
in a semicircle a right angle invariably (dafloAou) ? Because, if there be three 
straight lines, two forming tkt base, and the third iet uf at right angles at its 
middle point, the fact is obvious by simple inspection to any one who knows 
the property referred to" {Ikuvo is the property that the angles of a triangle 
are together equal to two right angles, mentioned two 
lines before). That is to say, the an^le at the middle 
point of the circumference of the semicircle was taken 
and proved, by means of the two isosceles right-angled 
triangles, to be the sum of two angles each equal to 
one-fourth of the sum of the angles of the large triangle 
in the figure, or of two right angles; and the proof . ?. -'.t 

must have been completed by means of the theorem of lit. zi (that angles 





£t BOOK III [III. 31 

in the same s^ment are equal), which Euclid's more general proof does 
not need. 

In the Greek texts before that of August there is an alternative proof 
that the angle BAC (in a semicircle) is right. August and Heiberg rel^;ate 
it to an Appendix. 

" Since the angle AEC is double of the angle BAE (for it is equal to the 
two interior and opposite angles), while the angle AEB b also double of the 
angle EAC, 

the angles AEB, AECatft double of the angle SAC. ■ ■ 

But the angles AEB, AEC are equal to two right angles J • ■• ' 1 
therefore the angle BAC is right." 

Lardner gives a slightly different proof of the second part of the theorem. 
If ABC be a segment greater than a semicircle, 
draw the diameter AD^ and join CH, CA. 

Then, in the triangle ACD, the angle ACD is right 
(being the angle in a semicircle) ; 

therefore the angle ADC\s acute. 
But the angle ADC i^ equal to the angle ABC in 
the same segment ; 

therefore the angle ABC b acute. 

Euclid's references in this proposition to the angle of a s^ment greater 
or less than a semicircle respectively seem, like the part of 111. 16 relating to 
the angle 0/ a semicircle, to be a survival of ancient controversies and not to 
be put in deliberately as being an essential part of elementary geometry. Cf. 
the notes on 111, Def. 7 and in. 16. 

The corollary ordinarily attached to this proposition is omitted by Heibetg 
as an interpolation of date later than Theon. It is to this effect ; " From 
this it is manifest that, if one angle of a triangle be equal to the other two, 
the first angle is right because the exterior angle to it is also equal to the 
same angles, and if the adjacent angles be equal, they are right." No doubt 
the corollary is rightly suspected, because there is no necessity for it here, and 
the words oirip itti Siifai come before it, not after tt, as is usual with Euclid. 
But, on the other hand, as the fact stated does appear in the proof of 111. 31, 
the Porism would be a Porism after the usual type, and I do not quite follow 
Hei berg's argument that, "if Euclid had wished to add it, he ought to have 
placed It after 1. 3*." 

It has already been mentioned above (p. 44) that this proposition supplies 
us with an alternative construction for the problem in 111, 1 7 of drawing the 
two tangents to a circle from an external point. 

Two theorems of some historical interest which follow directly from in. 3r 
may be mentioned. 

The first is a lemma of Pappus on " the 
14th problem " of the second Book of Apol- 
lonius' lost treatise on vcvVtit (Pappus vii. 
p. 811) and is to this effect. If a circle, as 
DEF, pass through D, the centre of a circle 
ABC, and if through F, the other point in 
which the line of centres meets the circle 
DEF, any straight line be drawn (and produced 
if necessary) meeting the circle DEF in E and the circle ABC in B, G, 




m- 3*. 3'] 



PROPOSITIONS 3t, %i 



«S 



then E is the middle point of £G. For, if UE be joined, the angle I>EJ^ 
(in a. semicircle) is a right angle [iii. 31] ; and DE, being at right angles to 
the chord BG of the circle A£C, also bisects it [m. 3]. 

The second is a proposition in the Zi^r Asiumpiarttm, attributed (no 
doubt erroneously as regards much of it) to Archimedes, which has reached 
us through the Arabic (Archimedes, ed, Heiberg, 11. pp. 52© — -5 21)' 

If two chords AB, CD iit a circle infersicl at right angles in a point O, 
thtn the sum of the squares on AG, BO, CO, DO is equal to the square on the 
diameter. 

For draw the diameter CE, and join AC, CB, AD, BE. , jj 




Then the angle CAO is equal to the angle CES. (This follows, in the 
first figure, from iii. 31 and, in the second, from 1. 13 and ill. 22.) Also the 
angle COA, being right, is equal to the angle CBE which, being the angle in a 
semicircle, is also right [iii. 31]. 

Therefore the triangles AOC, EBCh^ve two angles equal respectively; 
whence the third angles A CO, £CJ5 are equal. (In the second figure the 
angle A CO is, by i. 13 and 111. aa, equal to the angle ABD, and therefore 
the angles ABD, ECB are equal) 

Therefore, in both figures, the arcs AD, BE, and consequently the chords 
AD, BE subtended by them, are equal. [111. 36, 29] 

Now the squares on AO, DO are equal to the square on AD\\. 47}, that 
is, to the square on BE. 

And the squares on CO, £0 Mt equal to the square on BC. 

Therefore, by addition, the squares on AO, BO, CO, DO are equal to the 
squares on EB, BC, i.e. to the square on CE, [1. 47J 



Proposition 32. • ' 

If a straight line touch a circle, and from the point of 
contact there be drawn across, in the circle, a straight line 
cutting the circle, the angles which it makes with the tangent 
will be equal to the angles in the alternate segments of the 
circle. 

For let a straight line EF touch the circle A BCD at 
the point B, and from the point B let there be drawn across, 
in the circle ABCD, a straight line BD cutting it ; 
I say that the angles which BD makes with the tangent EF 
will be equal to the angles in the alternate segments of the 




M BOOK III [til. 31 

circle, that is, that the angle FBD is equal to the angle 
constructed in the segment BAD, and the angle EBD is 
equal to the angle constructed iii the 
segment DCB. 

For let BA be drawn from B at 
right angles to EF, 
let a point C be taken at random on 
the circumference BD, 
and let AD, DC, CB be joined. 

Then, since a straight line EF 
touches the circle A BCD at B, 
and BA has been drawn from the point 
of contact at right angles to the tangent, 
the centre of the circle ABCD is on BA. [m. 19] 

Therefore BA is a diameter of the circle ABCD ; 

therefore the angle ADB, being an angle in a semicircle, 
is right. [ill. 31] 

Therefore the remaining angles BAD, ABD are equal to 
one right angle. [1. 32] 

But the angle ABF is also right ; 
therefore the angle ABF is equal to the angles BAD, ABD. 

Let the angle ABD be subtracted from each ; 
therefore the angle DBF which remains is equal to the angle 
BAD in the alternate segment of the circle. 

Next, since ABCD is a quadrilateral in a circle, 
its opposite angles are equal to two right angles. [iii. a»] 

But the angles DBF, DBF are also equal to two right 
angles ; 

therefore the angles DBF, DBF are equal to the angles 
BAD, BCD, 

of which the angle BAD was proved equal to the angle 
DBF; 

therefore the angle DBF which remains is equal to the 
angle DCB in the alternate segment DCB of the circle. 

Therefore etc. q. e. d. 

The converse of this theorem is true, namely that, If a straight iine 
drawn through one txtraniiy of a chord of a circle make with that chord 
angles equal respectively to the angles in the alternate segments of the ctrde, 
the straight line so drawn touches the circle. 



""■ ih 33] 



PROPOSITIONS 33, 33 



67 



This can, as Camerer and Tod hunter remark, be proved indirectly ; or we 
may prove it, with Clavius, directly. Let BD be the given chord, and let £F 
be drawn through B so that it makes with BD angles equal to the angles in 
the alternate segments of the circle respectively. 

Let BA be the diameter through B, and let C be any point on the 
circumference of the segment DCB which does not contain A. Job A£>f 
DC, CB. 

Then, since, by hypothesis, the angle FBD is equal to the angle BAD, 
let the angle ABI> be added to both; 

therefore the angle ABF\i equal to the angles AJSD, BAD. 

But the angle BDA, being the angle in a semicircle, is a right angle ; 

therefore the remaining angles ABD, BAD in the triangle ABD are 
equal to a right angle. 

Therefore the angle ABFi^ right ; 
hence, since BA is the diameter through B, 

£i^ touches the circle at A [cii. 16, Por,] 

Pappus assumes in one place (iv, p. 196) the consequence of this 
proposition that, If two eircks touch, any straight line drawn through the point 
of (oniad and terminated by both cireiei mis off segments in each which are 
respediwly similar. Pappus also shows how to prove this (vii, p, 8i6) by 
drawing the cominon tangent at the point of contact and using thb proposition. 



Proposition 33. 

On a given straight line to describe a segment of a circle 
admitting an angle equal to a given rectilineal angle. 

Let AB be the given straight line, and the angle at C the 
given rectilineal angle; . . rv - 

thus it is required to describe 
on the given straight line 
AB a segment of a circle ad- 
mitting an angle equal to the 
angle at C. 

The angle at C is then 
acute, or right, or obtuse. 

First let it be acute, 
and, as in the first figure, on 
the straight line AB, and at the point A, let the angle BAD 
be constructed equal to the angle at C ; 

therefore the angle BAD is also acute. 

Let AE h& drawn at right angles to DA, let AB be 



\ 




«» 



rf BOOK m 



[""■ iS 



bisected at /^, let FG be drawn from the point F at right 
angles to AB, and let GB be joined. 

Then, since Af is equal to fB, 
and FG is common, 

the two sides AF, FG are equal to the two sides BF, FG ; 
and the angle AFG is equal to the angle BFG ; 

therefore the base AG is equal to the base BG. [i. 4] 

Therefore the circle described with centre G and distance 
GA will pass through B also. 

Let it be drawn, and let it be ABE ; 
let EB be joined. 

Now, since AD is drawn from A, the extremity of the 
diameter AE, at right angles to AE, 

therefore AD touches the circle ABE. \\\\. 16, Por.] 

Since then a straight line AD touches the circle ABE, 
and from the point of contact at A a straight line AB is 
drawn across in the circle ABE, 

the angle DAB is equal to the angle AEB in the alternate 
segment of the circle. [ni. 31] 

But the angle DAB is equal to the angle at C; 
therefore the angle at C is also equal to the angle AEB. 

Therefore on the given straight line AB the segment 
AEB of a circle has been described admitting the angle AEB 
equal to the given angle, the angle at C 1 ■ 

Next let the angle at C be right ; 




and let it be again required to describe on AB a segment 
of a circle admitting an angle equal to the right angle at C. 

Let the angle BAD be constructed equal to the right 
angle at C, as is the case in the second figure ; 



111. 33] PROPOSITION 33 69 

let j4B h& bisected at /^, and with centre /^ and distance 
either I^A or /^B let the circle AEB be described. 

Therefore the straight line AD touches the circle ABE, 
because the angle at A is right. [m. i6j Por.] 

And the angle BAD is equal to the angle in the segment 
AEB, for the latter too is itself a right angle, being an 
angle in a semicircle. [ni, 31) 

But the angle BAD is also equal to the angle at C. 

Therefore the angle AEB is also equal to the angle at C. 

Therefore again the segment AEB of a circle has been 
described on AB admitting an angle equal to the angle at C. 

Next, let the angle at C be obtuse ; 



.i»' 




"E . 'I 

and on the straight line AB, and at the point A, let the 
angle BAD be constructed equal to it, as is the case in the 
third figure ; 

let AE be drawn at right angles to AD, let AB be again 
bisected at F, let FG be drawn at right angles to AB, and 
let GB be joined. 

Then, since AF is again equal to FB, 
and FG is common, 

the two sides AF, FG are equal to the two sides BF, FG ; 
and the angle AFG is equal to the angle BFG ; 

therefore the base AG is equal to the base BG. [i- 4] 

Therefore the circle described with centre G and distance 
GA will pass through B also ; let it so pass, as AEB. 

Now, since AD is drawn at right angles to the diameter 
AE from its extremity, 

AD touches the circle AEB. [m. 16, Por.] 

And AB has been drawn across from the point of contact 
at W ; 

therefore the angle BAD is equal to the angle constructed 
in the alternate segment AHB of the circle. [m. 31] 



f0 BOOK in ' ["'33.34 

But the angle BAD is equal to the angle at C 
Therefore the angle in the segment A MB is also equal to 

the angle at C: 

Therefore on the given straight line AB the segment 

AHB of a circle has been described admitting an angle equal 

to the angle at C. 

Q, E, F. 

Simson remarks truly that the first and third cases, those namely in which 
the given angle is acute and obtuse respectively, have exactly the same 
construction and demonstration, so that there is no advantage in repeating 
them. Accordingly he deals with the cases as one, merely drawing two 
different figures. It is also true, as Simson says, that the demonstration of 
the second case in which the gi^-en angle is a right angle " is done in a round- 
about way," whereas, as Clavius showed, the problem can be more easily 
solved by merely bisecting AB and describing a semicircle on it. A glance 
at Euclid's figure and proof will however show a more curious fact, namely 
that he does not, in the proof of the second case, use the angle in the 
alternate stgmint, as he does in the other two cases. He might have done so 
after proving that AD touches the circle; this would only have required his 
point .£ to be placed on the side of AB opposite to D. Instead of this, he 
uses III. 31, and proves that the angle AEB is equal to the angle C, because 
the former is an angle in a sanicirde, and is therefore a right angle as C is. 

The difference of procedure is no doubt owing to the fact that he has not, 
in III. 31, distinguished the case in which the cutting and touching straight 
lines are at right angles, i.e. in which the two alternate segments are semicircles. 
To prove this case would also have required in. 31, so that nothing would 
have been gained by stating it separately in in. 32 and then quoting the 
result as part of 111. 32, instead of referring directly to in. 31. 

It is assumed in Euclid's proof of the first and third cases that AE and 
FG will meet; but of course there is no difficulty in satisfying ourselves 
of this. J 



Proposition 34. "" " " 

From a given circle to cut off a segment admitting an angle 
tqual to a given rectilineal angle. 

Let ABC be the given circle, and the angle at D the 
given rectilineal angle ; 

thus it is required to cut off from the circle ABC a segment 
admitting an angle equal to the given rectilineal angle, the 
angle at D. 

Let EF\i^ drawn touching ABC at the point B, and on 
the straight line FB, and at the point B on it, let the angle 
FBC be constructed equal to the angle at D. [1. 23] 

' . Then, since a straight line EF touches the circle ABC^ 



"■•34. 3S] PROPOSITIONS 33-35 ji 

and BC has been drawn across from the point of contact 
at^, 

the angle FBC is equal to the angle constructed in the alternate 
segment BAC, [iti. 37] 



But the angle FBC is equal to the angle at D ; 

therefore the angle in the segment BAC is equal to the 
angle at D. 

Therefore from the given circle ABC the segment BAC, 
has been cut off" admitting an angle equal to the given recti- 
lineal angle, the angle at D. 

Q. E, F. 

An alternative construction here would be to make an "angle at the 
centre " {in the extended sense, if necessary) double of the given angle ; and, 
if the given angle is right, it is only necessary to draw a diameter of the circle. 



Proposition 35. 

Jf in a circle two straight lines cut one another, the 
rectangle contained by the segments of the one is equal to the 
rectangle contained by the segments of the other. 

For in the circle ABCD let the two straight lines AC, 
BD cut one another at the point E ; 

I say that the rectangle contained hy AB, 
EC is equal to the rectangle contained by 
DE, EB. 

If now AC, BD are through the centre, 
so that E is the centre of the circle ABCD, 

it is manifest that, AE, EC, DE, EB 
being equal, 

the rectangle contained by AE, EC is also equal to the 
rectangle contained by DE, EB. 





:ja /; BOOK in [hi. 35 

Next let AC, DB not be through the centre ; 
let the centre of ABCD be taken, and 
let it be F\ 

from F let FG, FH be drawn perpen- 
dicular to the straight lines AC, DB, 
and let FB, FC, FE be joined. 

Then, since a straight line GF 
through the centre cuts a straight line 
AC not through the centre at right 
angles, 

it also bisects it ; [in. 3] 

therefore AG is equal to GC. 

Since, then, the straight line AC has been cut into equal 
parts at G and into unequal parts at E, 

the rectangle contained by AE, EC together with the square 
on EG is equal to the square on GC ; [11. 5] 

Let the square on GF be added ; 
therefore the rectangle AE, EC together with the squares 
on GE, GF is equal to the squares on CG, GF. 

But the square on FE is equal to the squares on EG, GF, 
and the square on FC Is equal to the squares on CG, GF\ 

['. 47l 

therefore the rectangle AE, EC together with the square 
on FE is equal to the square on FC. 

And FC is equal to FB ; 
therefore the rectangle AE, EC together with the square on 
EF is equal to the square on FB. 

For the same reason, also, 
the rectangle DE, EB together with the square on FE is 
equal to the square on FB. 

But the rectangle AE, EC together with the square on 
FE was also proved equal to the square on FB ; 
therefore the rectangle AE, EC together with the square on 
FE is equal to the rectangle DE, EB together with the 
square on FE. 

Let the square on FE be subtracted from each ; 
therefore the rectangle contained by AE, EC which remains 
is equal to the rectangle contained by DE, EB, 

Therefore etc. 



III. 35, 36] PROPOSITIONS 35. 36 m 

In addition to the two cases in Euclid's text, Simson (following Campanus) 
gi.ves two intermediate cases, namely (i) that in which one chord passes through 
the centre and bisects the other which does not pass through the centre at right 
angles, and (a) that in which one passes through the centre and cuts the other 
which does not pass through the centre but not at right angles Simson then 
reduces Euclid's second case, the most general one, to the second of the two 
intermediate cases by drawing the diameter through £. His note is as 
follows : "As the 25th and 33rd propositions are divided into more cases, 
so this 35th is divided into fewer cases than are necessary. Nor can it be 
supposed that Euclid omitted them because they are easy ; as he has given 
the case which by far is the easiest of them all, viz, that in which both the 
straight lines pass through the centre ; And in the following proposition he 
separately demonstrates the case in which the straight line passes through the 
centre, and that in which it does not pass through the centre: So that it 
seems Theon, or some other, has thought them too long to insert : But cases 
that require different demonstrations should not be left out in the Elements, 
as was before taken notice of: These cases are in the translation from the 
Arabic and are now put into the text." Notwithstanding the ingenuity of the 
argument based on the separate mention by Euclid of the simplest case of 
all, I think the conclusion that Euclid himself gave /our cases is unsafe ; in 
fact, in giving the simplest and most difficult cases only, he seems to be 
following quite consistently his habit of avoiding Aw ^reai multiplicity of cases, 
while not ignoring their existence. 

The deduction from the next proposition (in, 36) which Simson, following 
Clavius and others, gives as a corollary to it, namely that, IJ from any point 
without a drcU then be drawn two straight tines cutting it, the rectangles 
contained by the whole lines and the parts of them without the circle are equal t& 
one another, can of course be combined with ill. 35 in one enunciation. 

As remarked by Todhunter, a large portion of the proofs of in, 35, 36 
amounts to proving the proposition, If any point be taken on the bast, or the 
base produced, of an isosceles triangle, the rectangle contained by the segments of 
the base (i.e. the respective distances of the ends of the base from the point) is 
equal to the difference betiveen the square on the straight line joining the point to 
the vertex and the square on one of the equal sides of the triangle. This is of 
course an immediate consequence of 1, 47 combined with ii. 5 or 11. 6, 

The converse of in, 35 and Simson's corollary to lu. 36 may be stated 
thus. If two straight lines AB, CXi, produced if necessary, intersect at O, and if 
the rectangle AO, OB be equal to the rectangle CO, OD, the circumference of a 
circle will pass through the four points A, B, C, D. The proof is indirect. 
We describe a circle through three of the points, as A, B, C (by the method 
used in Euclid's proofs of tii. 9, 10), and then we prove, by the aid of in. 35 
and the corollary to in. 36, that the circle cannot but pass through D also, 

» . .ii\ .<i A TM 

•.' ^. ■ 5 ■■- . 'fi v'* r •(' I 

Proposition 36. 

If a point be taken outside a circle and from it there fall 
on the circle two straight lines, and if one of tliem cut the 
circle and the other touch ii, the rectangle contained by the 
whole of the straight line which cuts the circle and the straight 




f4 BOOK, in ■: [111.36 

line intercepted on it outside between the point and the convex 
circumference will be equal to the square on tke tangent. 

For let a point D be taken outside the circle ABC, 
and from D let the two straight lines DC A, 
DB fall on the circle ABC; let DCA cut 
the circle ABC and let BD touch it ; 
I say that the rectangle contained by AD, 
DC is equal to the square on DB. 

Then DCA is either through the centre 
or not through the centre. 

First let it be through the centre, and 
let F be the centre of the circle ABC; 
let FB be joined ; 

therefore the angle FBD is right. [m. 18] 

And, since AC has been bisected at F, and CD is added 
to it, 

the rectangle AD, DC together with the square on FC is 
equal to the square on FD. [11. 6] 

But FC is equal to FB ; 
therefore the rectangle AD, DC together with the square on 
FB is equal to the square on FD. 

And the squares on FB, BD are equal to the square on 
FD ; [i. 47] 

therefore the rectangle AD, DC together with the square on 
FB is equal to the squares on FB, BD. 

Let the square on FB be subtracted from each ; 
therefore the rectangle AD, DC which remains is equal to 
the square on the tangent DB. 

Again, let DCA not be through the centre of the circle 
ABC; 

let the centre E be taken, and from E 
let EF be drawn perpendicular x.o AC; 
let EB, EC, ED be joined. 

Then the angle EBD is right. 

[ill. 18] 
And, since a straight line EF 
through the centre cuts a straight line 
AC not through the centre at right angles, 

it also bisects it ; [in. 3] 

therefore AF is equal to FC. 




jii. 36. 37] PROPOSITIONS 36, 37 ?S 

Now, since the straight line ^Chas been bisected at the 
point F, and CD is added to it, 

the rectangle contained by AD, DC together with the square 
on FC is equal to the square on FD. [11, 6] 

Let the square on FE be added to each ; 
therefore the rectangle AD, DC together with the squares 
on CF, FE is equal to the squares on FD, FE. 

But the square on EC is equal to the squares on CF, FE, 
for the angle EEC is right ; [1. 47] 

and the square on ED is equal to the squares on DF, FE ; 
therefore the rectangle AD, DC together with the square on 
EC is equal to the square on ED. 

And EC is equal to EB ; 
therefore the rectangle AD^ DC together with the square on 
EB is equal to the square on ED. 

But the squares on EB, BD are equal to the square on 
ED, for the angle EBD is right ; [i. 47] 

therefore the rectangle AD, DC together with the square on 
EB is equal to the squares on EB, BD. 

Let the square on EB be subtracted from each ; 
therefore the rectangle AD, DC which remains is equal to 
the square on DB. 

Therefore etc, q. e. d, 

Cf. note on the preceding proposition. Observe that, whereas it would 
be natural with us to prove first that, if A is an external point, and two 
straight lines AEB, AFC cut the circle in E, B and F, C respectively, the 
rectangle BA, AE h equal to the rectangle CA, AF, and thence ihat, the 
tangent from A being a straight tine likt AEB in its limiting position when 
E and B coincide, either rectangle is equal to the square on the tangent 
(cf. Mr H. M. Taylor, p, 153), Euclid and the Greek geometers generally did 
not allow themselves to infer the truth of a proposition in a limiting case 
directly from the general case including it, but preferred a separate proof of 
the limiting case (cf. Apollonius of Perga, p. 40, 139 — 140). This accounts for 
the form of 11 r. 36. 



PRorosiTioN 37, 

If a point be taken outside a iireie and from the point 
there fall on the circle two straight lines, if one of them cut 
the circle, and the other fall on it, and if further the rect- 
angle contained by tlte whole of the straight line which cuts 




^1 ,. BOOK III [hi, 37 

Ike circle and the straight line intercepted on it outside 
between the point and the convex circumference be equal to 
the square on the straight line which falls on the circle, the 
straight line which falls on it will touch the circle. 

For let a point D be taken outside the circle ABC; 
from D let the two straight lines 
DCA, DB fall on the circle ACB; 
let DC A cut the circle and DB 
fall on it ; and let the rectangle AD, 
DC be equal to the square on DB. 

I say that DB touches the circle 
ABC. 

For let DE be drawn touching 
ABC ; let the centre of the circle ABC be taken, and let it 
be F\ let FE, FB, FD be joined. 

Thus the angle FED is right. [m. i8] 

' ' Now, since DE touches the circle ABC, and DC A cuts it, 
the rectangle AD, DC is equal to the square on DE, [m. 36] 

But the rectangle AD, DC vt^.s also equal to the square 
onDB; 
therefore the square on DE is equal to the square on DB ; 

therefore DE is equal to DB. 
■ And FE is equal to FB ; 1 

therefore the two sides DE, EF are equal to the two sides 
DB, BF; 
and FD is the common base of the triangles ; 

therefore the angle DEF is equal to the angle DBF. 

[l 8] 
But the angle DEF is right ; 

therefore the angle DBF is also right. 
And FB produced is a diameter ; 
and the straight line drawn at right angles to the diameter 
of a circle, from its extremity, toucnes the circle ; [iir. 16, For.] 
therefore DB touches the circle. 
Similarly this can be proved to be the case even if the 
centre be on ^C. 1 

Therefore etc. • <- q. e. d. ' 

De Morgan observes that there is here the same defect as in i. 48, i.e. an 
apparent avoidance of indirect demonstration by drawing the tangent DE on 



Iir, 37] PROPOSITION 37 ft 

the 0(^)OSite side of DF from UB. The case is similar to the appartnily 
direct proof which Campanus gave. He drew the straight line from D 
passing through the centre, and then (without drawing a second tangent) 
proved by the aid of n. 6 that the square on DP is equal to the sum of the 
squares on DB, BF\ whence {by t. 48) the angle DBF is a right angle. 
But this proof uses I. 48, the very proposition to which De Morgan's original 
remark relates. 

The undisguised indirect proof is easy. If DB does not touch the circle, 
it must cut it if produced, and it follows that the square on DB must be 
equal to the rectangle contained by DB and a longer line ; which is absurd. 



;ti 



' t 



BOOK IV. 

DEFINITIONS. 

I. A rectilineal Bgure is said to be inscribed in a 
rectilineal figure when the respective angles of the 
inscribed figure lie on the respective sides of that in which 
it is inscribed. 

i. Similarly a figure is said to be circumscribed about 
a figure when the respective sides of the circumscribed 
figure pass through the respective angles of that about which 
it is circumscribed. 

3. A rectilineal figure is said to be inscribed in a 
circle when each angle of the inscribed figure lies on the 
circumference of the circle. 

4. A rectilineal figure is said to be circumscribed 
about a circle, when each side of the circumscribed figure 
touches the circumference of the circle. 

5. Similarly a circle is said to be inscribed in a figure 
when the circumference of the circle touches each side of the 
figure in which it is inscribed. 

6. A circle is said to be circumscribed about a figure 
when the circumference of the circle passes through each 
angle of the figure about which it is circumscribed. 

7. A straight line is said to be fitted into a circle when 
its extremities are on the circumference of the circle. 

Definitions 1—7. ■ 

I api>end, as usual, the Greek text of the definitions. 

I, ^xi^jxa (v$vypafjLfiov tU (T^/lii tddvypafiftov tyypdifntTdat AryCTCU, orAy 
ttaimj TiSv Tol fyypa^fttKOu o^jtaTOt ymyitar iiida-rft wXtvpa^ rou, tit S 
iyypa^rTtLif airnfrai. 



IV, DEFF-. a— 7] DEFINITIONS 1—7 n 

5, 2;^/xa lit^uypofifiav tl^ kvkXov Iffpo^^vStui Xtycrac, &rav cffairn^ ytaviti 
ToS tffpat^jiivav aiTTTfrat T^s tow kvicXod wtpi^tpiia.%. 

tXcitu tdS wtpiypa^ftivov i^Tmjrai riji tou kvkXou vtpi^tptiat. 

5. KvitXot j4 (tt cr;(i7/ta o/UKOff lyypd()>i<rBai \iyirat, Srav 1) Tov kvkXou 

6. KujfXo^ M ir^pl vx^fjta TTfpiypd^trSai Xrycrac, Jrav 17 tov ^icXov iripi^ipiia 
iitwmj^ 'yw»'(a¥ tou, wtpi S vtpiyp^^Tat, aTmjrtii. 

7. Eutftra (ft KuKXer Jra/ijufffO'dai Xrytrai, oral' tu irc^aTa avr^f JTrt t^s 
wtptf^ptuL^ ^ Toi) fcuxXou, 

In the Rrst two definitions an English translation, if il is to be clear, must 
depart slightly from the exact words used in the Greek, where "each side" of 
one figure is said to pass through " each angle " of another, or " each angle " 
{I.e. angular point) of one ties on " each side " of another {Udimi rktvpd, 
iiiaiTD/ ytiiyia). 

It is also necessary, in the five definitions 1, 1, 3, 5 and 6, to translate 
the same Greek word aTrr^Toi in three different ways. It was observed on 
tit. Def. 2 that the usual meaning of arrurSai in Euclid is to metf, in contra- 
distinction to li^Trrt<r9ai, which means to fau^A. Exceptionally, as in Def. 5, 
iimaSoi has the meaning of iaucA. But two new meanings of the word appear, 
the first being to /ie on, zs in DefT. i atid 3, the second to pass through, as in 
DefT. 3 and 6; "each angle" lies on (airriTat) a side or on a circle, and 
" each side," or a circle, passes through (aurrrai) an angle or " each angle," 
The first meaning of lying an is exemplified in the phrase of Pappus af ctoi tJ 
a)\pM.w $(cr<( hi%op^lyrrp (Mtiat, "will lie on a Straight line given in position"; 
the meaning of passing through seems to be much rarer (I have not seen it in 
Archimedes or Pappus), but, as pointed out on itt. Def. 2, Aristotle uses the 
compound l^-miahai. in this sense. 

Simson proposed to read i^imjrat in the case (Def. 5) where an-njTot 
means touches. He made the like suggestion as regards the Greek text of ttl. 
II, 12, ij, 18, 19; in the first four of these cases there seems to be ms. 
authority for the compound verb, and in the fifth He! berg adopts Slmson's 
correction. 



,[■ <Mn-' .'1 », . ! • «! 

■'«';' . I- .J ^ •,.•?: ,.. .^ <{".<■ -'•to •y..\:z ,.T; n ' 



.•1 



BOOK IV. PROPOSITIONS 



PrOI'OSITION I. 




fn/o a £iven circle to fit a straight line equal to a given 
straight line which is not greater than the diameter of the 
circle. 

Let ABC be the given circle, and D the given straight 
line not greater than the diameter 
of the circle ; 

thus it is required to fit into the 
circle ABC a straight line equal 
to the straight line D. 

Let a diameter BC of the 
circle ABC be drawn. 

Then, if BC is equal to D, 
that which was enjoined will have 

been done ; for BC has been fitted into the circle ABC equal 
to the straight line D. 

But, if BC is greater than D, 

let CE be made equal to D, and with centre C and distance 
CE let the circle EAR be described ; 

let CA be joined. 

Then, since the point C is the centre of the circle EAF, 

CA is equal to CE. 

But CE is equal to /? ; 

therefore D is also equal to CA. 

Therefore into the given circle ABC there has been fitted 
CA equal to the given straight line D. 



IV. I, a] PROPOSITIONS i, i $i 

or this problem as it stands there are of course an infinite number of 
solutions; and, if a particular point be chosen as one extremity of the chord 
to be "fitted in," there are two solutions. More difficult cases of "fitting 
into " a circle a chord of given length are arrived at by adding some further 
condition, e.g. (i) that the chord is to be parallel to a given straight line, or 
(2) that the chord, produced if necessary, shall pass through a given point. 
The former problem is solved by Pappus (in. p. rja); instead of drawing the 
chord as a tangent to a circle concentric with the given circle and having as 
radius a straight line the square on which is equal to the difference between 
the squares on the radius of the given circle and on half the given length, he 
merely draws the diameter of the circle which is parallel to the given direction, 
measures from the centre along it in each direction a length equal to half the 
given length, and then draws, on one side of the diameter, perpendiculars to it 
through the two points so determined. 

The second problem of drawing a chord of given length, being less than 
the diameter of the circle, and passing through a given point, is more 
important as having been one of the problems discussed by Apollonius in his 
work entitled vnxriit, now lost. Pappus states the problem thus (vii. p. 670): 
"A circle being given in position, to fit into it a straight line given in 
magnitude and verging (vtvoixrac) towards a given (point)." To do this we 
have only to place any chord HK in the given 
circle (with centre O) equal to the given length, 
take Z the middle point of it, with O as centre and 
OL as radius describe a circle, and lastly through 
the given point C draw a tangent to this circle 
meeting the given circle in j4, B, AB is then one 
of two chords which can be drawn satbfying the 
given conditions, if C is outside the inner circle ; if 
C\%Bn the inner circle^ there is one solution only ; 
and, if C is within the inner circle, there is no 
solution. Thus, if C is within the outer (given) 

circle, besides the condition that the given length must not be greater than the 
diameter of the circle, there is another necessary condition of the possibility 
of a solution, viz. that the given length must not be Itss than double of the 
straight line the square on which is equal to the difference between the squares 
(i) on the radius of the given circle and (2) on the distance between its 
centre and the given point. 



Proposition 2. 

In a given circle to inscribe a triangle equiangular with a 
given triangle. 

Let ABC be the given circle, and DBF the given 
triangle ; 

thus it is required to inscribe in the circle ABC a triangle 
equiangular with the triangle DEF. 

Let GH\x. drawn touching the circle ABC at A [m. i6,Por.]; 




8a BOOK TV • [iT. » 

on the straight line AH, and at the point A on it, let the 
angle HAC be constructed eoual to the angle DEF, 
and on the straight line AG, and at the point A on it, let 
the angle GAB be constructed equal to the angle DFE ; 

let BC be joined. 




Then, since a straight line AH touches the circle ABC, 
and from the point of contact at A the straight line ^C is 
drawn across in the circle, 

therefore the angle HA C is equal to the angle ABC in the 
alternate segment of the circle. fin. 3*] 

But the angle HA C is equal to the angle DEF ; 
therefore the angle ABC is also equal to the angle DEF. 

For the same reason 

the angle ACB is also equal to the angle DFE ; 
therefore the remaining angle BAC is also equal to the 
remaining angle EDF. [i. 3*] 

Therefore in the given circle there has been inscribed a 
triangle equiangular with the given triangle. q. e. f. 

Here again, since any point on the circle niay be taken as an angular 
point of the triangle, there are an infinite number of solutions. Even when a 
particular point has been chosen to form one angular point, the required 
triangle may be constructed in six ways. For any one of the three angles 
may be placed at the point ; and, whichever is placed there, the positions of 
the two others relatively to it may be interchanged. The sides of the triangle 
will, in all the different solutions, be of the same length respectively ; only 
their relative positions will be different 

This problem can of course be reduced (as it was by Borelli) to nt. 34, 
namely the problem of cutting off from a given circle a segment containing an 
angle equal to a given angle. It can also be solved by the alternative method 
applicable to ni. 34 of drawing " angles at the centre " equal to double the 
angles of the given triangle respectively ; and by this method we can easily 
solve this problem, or \\\. 34, with the further condition that one aide of the 



IV. 3, 3] PROPOSITIONS a, 3 Sj 

required triangle, or the base of the required segment, respectively, shall be 
parallel to a given straight line. 

As a particular case, we can, by the method of this proposition, describe 
an tguilaterai triangle in any circle after we have first constructed any 
equilateral triangle by the aid of i. i. The possibility of this is assumed in 
IV. t6. It is of course equivalent to dividing the circumference of a circle 
into I Arte equttl parti. As De Morgan says, the idea of dividing a revolution 
into equal parts should be kept prominent in considering Book iv. ; this 
aspect of the construction of regular polygons is obvious enough, and the 
reason why the division of the circle into fh-et equal parts is not given by 
Euclid is that it happens to be as easy to divide the circle into three parts 
which are in the ratio of the angles of any triangle as to divide it into three 
equal parts. 

Proposition 3. 

About a given circle to circumscribe a triangle equiangular 
with a given triangle. 

Let ABC be the given circle, and DEF the given 
triangle ; 

1 thus it is required to circumscribe about the circle ABC a 
triangle equiangular with the triangle DEF. 




Let EF be produced in both directions to the points 
G, H, 
let the centre K of the circle ABC be taken [in. r], and let 

10 the straight line KB be drawn across at random ; 
on the straight line KB^ and at the point K on it, let the 
angle SKA be constructed equal to the angle DEG, 
and the angle BKC equal to the angle DFH ; [i. J3] 

and through the points A, B, C let LAM, MEN, NCL be 

15 drawn touching the circle ABC. [in. 16, For] 

Now, since LM, MN, NL touch the circle ABC at the 
points A, B, C, 

and KA, KB, KC have been joined from the centre K to 
the points A, B, C, 



S| - BOOK IV ' itr>^ 

ao therefore the angles at the points A, B, C are right. [iii. i8] 

And, since the four angles of the quadrilateral AMBK 
are equal to four right angles, inasnnuch as AMBK is in fact 
divisible into two triangles, 

and the angles KAM, KBM are right, 

25 therefore the remaining angles A KB, A MB are equal to two 

right angles. 

But the angles DEG, DEF are also equal to two right 

angles ; [1. 13] 

therefore the angles A KB, A MB are equal to the angles 
30 DEG, DEF, 

of which the angle A KB is equal to the angle DEG ; 

therefore the angle AMB which remains is equal to the 
angle Z?^/^ which remains, 

Similarly it can be proved that the angle LNB is also 
3S equal to the angle DFE \ 

therefore the remaining angle MLN is equal to the 

angle EDF. \\. 33] 

Therefore the triangle LMN is equiangular with the 

triangle DEF; and it has been circumscribed about the 

40 circle ABC. 

Therefore about a given circle there has been circum- 
scribed a triangle equiangular with the given triangle. 

Q, E. F. 

ii». at raodom, Uterslly " ax it ma; chance," in trt/x"- The same etpression is used 
in ][]. I and commonly. 

11. Is In fact dlviaible, lol SuupttTai, literally " is actually divided." 

The remarks as to the number of ways in which Prop, a can be solved 
apply here also. 

Euclid leaves us to satisfy ourselves that the three tangents }t>t7i meet and 
form a triangle. This follows easily from the fact that each of the artgles 
A^B, BKC, CKA is less than two right angles. The first two are so by 
construction, being the supplements of two angles of the given triangle re- 
spectively, and, since ail three angles round K are together equal to four 
right angles, it follows that the third, the angle AKC, is equal to the sum 
of the two angles E, Foi the triangle, i.e. to the supplement of the angle D, 
and is therefore less than two right angles. 

Peletarius and Borelli gave an alternative solution, flrst inscribing a triangle 
equiangular to the given triangle, by iv. 2, and then drawing tangents to the 
circle parallel to the sides of the inscribed triangle respectively. This method 
will of course give two solutions, since two tangents can be drawn parallel to 
each of the sides of the inscribed triangle. 

If the three pairs of parallel tangents be drawn and produced far enough, 




IV. 3, 4] PROPOSITIONS s, 4 »S 

they will form ^I'^Af triangles, two of which are the triangles ctrcumKribed to 
the circle in the manner required in the proposition. The other six triangles 
are so related to the circle that the circle touches two of the sides in each 
produced, i.e. the circle is an escribed circle to each of the six triangles. 



Proposition 4. 
In a given triangle to inscribe a circle. 

Let ABC be the given triangle ; 
thus it is required to inscribe a circle in the triangle ABC. 
Let the angles ABC, ACB 
S be bisected by the straight Hnes 
BD, CD [1. 9], and let these meet 
one another at the point D ; 
from D let DE, DF, DG be 
drawn perpendicular tothestraight 
10 lines AB, BC, CA. 

Now, since the angle ABD 
is equal to the angle CBD, 

and the right angle BED is also equal to the right angle 
BFD, 
IS EBD, FBD are two triangles having two angles equal to two 
angles and one side equal to one side, namely that subtending 
one of the equal angles, which is BD common to the 
triangles ; 

therefore they will also have the remaining sides equal to 
20 the remaining sides ; [i. 36] 

therefore DE is equal to DF, 
For the same reason 

DG is also equal to DF. 
Therefore the three straight lines DE, DF, DG are ec;ual 
as to one another ; 

therefore the circle described with centre D and distance 

one of the straight lines DE, DF, DG will pass also 

through the remaining points, and will touch the straight 

lines AB, BC, CA, because the angles at the points E, F, G 

30 are right. 

For, if it cuts them, the straight line drawn at right angles 
to the diameter of the circle from its extremity will be found 
to fall within the circle : which was proved absurd ; [in. 16] 



■9$ . BOOK IV [iv. 4 

therefore the circle described with centre D and distance 
35 one of the straight lines £>£, BF, DG will not cut the 
straight lines AB, BC, CA ; 

therefore it will touch them, and will be the circle inscribed 
in the triangle ABC. ["v. Def. s] 

Let it be inscribed, as FGE. 
4° Therefore in the given triangle ABC the circle EFG has 
been inscribed. . - . ■.. ,. , , 

y. t,. r. 

i6, 34. and distance one of the (straight lines D)E, (D)F, (D)G. The wonte 
«nd letters here shown in brarkets are put in to fill out tbe ralher carelcK hngiiige o( ihe 
Greek- Hete and in several other places in Book IV. Euclid says lilemliy "and w I ih distance 
one of the (points) E,F,G" (lai Juunt^tan M rwc E, Z, H) and the like. In one case (1 v. 13) 
he actually has " with distance one of the pttintt G, If, IC, L, M" {tiuHmaTi irl rSr H, ©, 
K, A, H irit;i(lup). Heiberg notes" Craecam loculionem satis miram et negligenlem," but, 
in view of its frequent occurrence in good M3S., does not venture to correct it. 

Euclid does not think it necessary to prove that ££>, CD will meet ; this 
is indeed obvious, for the angles DEC, DCB are together half of the angles 
ABC, ACB, which themselves are tc^ether less than two right angles, and 
therefore the two bisectors of the angles B, C must meet, by Post. 5. 

It follows from the proof of this proposition that, if the bisectors of two 
angles B, C q( b. triangle meet in D, the line joining D ia A also bisects the 
third angle A, or the bisectors of the three angles. of a* triangle meet in 
a point 

It will be observed that Euclid uses the indirect form of proof when 
showing that the circle touches the three sides of the triangle. Simson proves 
it directly, and points out that Euclid does the same in 111. 17, 33 and 37, 
whereas in iv. 8 and 13 as well as here he uses the indirect form. The 
difference is unimportant, being one of fonn and not of substance; the 
indirect proof refers back to in. 16, whereas the direct refers back to the 
Porism to that proposition. 

We may state this problem in the moie general form : Te describe a circle 
touching three given straight lines which do net all meet in one point, and of 
which not mere than two are parallel. 

In the case (i) where two of the straight lines are parallel and the third 
cuts them, two pairs of interior angles are formed, one on each side of the 
third straight line. If we bisect each of the interior angles on one side, the 
bisectors will meet in a point, and this point will be the centre of a circle 
which can be drawn touching each of the three straight lines, its radius being 
the perpendicular from the point on any one of the three. Since the alternate 
angles are equal, two equal circles can be drawn in this manner satisfying the 
given condition. 

In the case (2) where the three straight lines form a triangle, suppose each 
straight line produced indefinitely. Then each straight line will make two 
pairs of interior angles with the other two, one pair forming two angles of the 
triangle, and the other pair being their supplements. By bisecting each angle 
of either pair we obtain, in the manner of the proposition, two circles 
satisfying the conditions, one of them tieing the inscribed circle of the triangSe 
and the other being a circle eseriied to it, i.e. touching one side and the other 



IV. 4] 



PROPOSITION 4 



two sides ptvduftd. Next, taking the pairs of interior angles formed by a 
second side with the other two produced indefinitely, we get two circles 
satisfying the conditions, one of which is the same inscribed circle that we had 
before, while the other is a second escribed circle. Similarly with the third side. 
Hence we have the inscribed circle, and three escribed circles (one opposite 
each angle of the triangle), i.e. four circles in all, satisfying the conditions of 
the probiem. 

it ntay perhaps not be inappropriate to give at this point Heron's elegant 
proof of the formula for the area of a triangle in terms of the sides, which we 
usually write thus : 

A=Js{f-a){s-b){s-c), 

although it requires the theory of proportions and uses some ungeometrical 
expressions, e.g. the product of two areas and the "side " of such a product, 
where of course the areas are so many square units of length. The proof is 
given in the Metrica, i. 8, and in the Dioptra, 30 (Heron, Vol. iii., Teubner, 
190J, pp. *o— i4 and pp. j8o — 4, or Heron, ed. Hultsch, pp. 235 — 7). 

Suppose the side-s of the triangle ABC\o be given in length. 

Inscribe the circle DEF, and let G be its centre. 




If ■! 



Join AG, BG, CG, DG, EG, FG. 

Then EC. EG =2. A BGC, 

CA. FG=i.£:.ACG, 

AB.DG=2.£.ABG. 

Therefore, by addition, 

p.EG^i.CiABC, 
where/ is the perimeter. 

Produce CB to H, so that BH^ AD. 
Then, since AD = AF, DB = BE, FC = CE, 

CH= y. 

Hence CH. EG=t, ABC. 



as BOOK IV [iv. 4, 5 

But CH .EG is the "side" of the product CH^ . EC, that b 
JCH^.EG^; 

therefore {i^ABC)^=Cir.EG\ 

Draw GL at right angles to CG, and BL at right angles to CB, meeting 
at L. Join CL. 

Then, since each of the angles CGL, CBL is right, CGBL is a quadri- 
lateral in a circle. 

Therefore the angles CGB, CLB are equal to two right angles. 
Now the angles CGB, AGD are equal to two right angles, since AG, BG, 
CG bisect the angles at G, and the angles CGB, AGD are equal to the 
angles AGC, DGB, while the sum of all four is equal to four right angles. 
Therefore the angles j4 CZJ, CZZf are equal. 
So are the right angles ADG, CBL. 
Therefore the triangles AGD, CLB are similar. 
Hence BC: BL = AD-.DG 

= BH: EG, 
and, alternately, CB : BH = BL : EG 

= BK: KE, 
whence, tomponende, CH: HB = BE : EK. 

It follows that CH-" : CH . HB ^ BE . EC.CE . EK 

'^ BE. EC: EG* 
Therefore 

(A ABC)^ = CH' . EG'= CH. HB . CE . EB 

^\P{\p-BC){\p-AB){\p-AO. 

Proposition 5. 

About a ^ven triangle to circumscribe a circle. 

Let ABC be the given triangle ; 
thus it is required to circumscribe a circle about the given 
triangle ABC. 






Let the straight lines AB, AC h& bisected at the points 
D, E [i. 10], and from the points D, E let DF, EF be drawn 
at right angles to AB, AC ; 

they will then meet within the triangle ABC^ or on the 
straight line BC, or outside BC. 



IV. 5j PROPOSITIONS 4, s 89 

First let them meet within at /% and let FB, FC, FA be 
joined. 

Then, since AD is equal to DB, ■ . f 

and DF is common and at right angles, 
therefore the base AFis equal to the base FB. [!■ 4] 

Similarly we can prove that 

CF is also equal to ^^; ' 

so that FB is also equal to FC ; 
^ therefore the three straight Hnes FA, FB, FC are equal 
to one another, 

Therefore the circle described with centre F and distance 
one of the straight lines FA, FB, FC will pass also through 
the remaining points, and the circle will have been circum- 
scribed about the triangle ABC. 

Let it be circumscribed, as ABC. 

Next, let DF, EF meet on the straight line BC at F, 
as is the case in the second figure ; and let AF be joined. 

Then, similarly, we shall prove that the point F is the 
centre of the circle circumscribed about the triangle ABC. 

Again, let DF, EF meet outside the triangle ABC at F, 
as is the case in the third figure, and let AF, EF, CF be 
joined. 

Then again, since AD is equal to DB, 

and DF'is common and at right angles, 

therefore the base AF is equal to the base BF. [i. 4] 

Similarly we can prove that 

C/^ is also equal to v^/^; 
so that BF is also equal to FC ; 

therefore the circle described with centre F and distance one 
of the straight lines FA, FB, FC will pass also through 
the remaining points, and will have been circumscribed about 
the triangle ABC. 

Therefore about the given triangle a circle has been 
circumscribed, 

Q. E. F. 

And it is manifest that, when the centre of the circle falls 
within the triangle, the angle BAC, being in a segment 
greater than the semicircle, is less than a right angle ; 



90 BOOK IV [iv. s 

when the centre falls on the straight line BC, the angle BAC, 
being in a semicircle, is right ; 

and when the centre of the circle falls outside the triangle, 
the angle BAC, being in a segment less than the semicircle, 
is greater than a right angle, [m. 31] 

Sim son points out that Euclid does not prove that DF, EFmW meet, and 
he inserts in the text the following argument to supply the omission. 

"^ DF, ^/i" produced meet one another. For, if they do not meet, they 
are parallel, wherefore AB, AC, which are at right angles to them, are 
parallel [or, he should have added, in a straight line] : which is absurd." 

This assumes, of course, that straight lines which are at right angles to two 
parallels are themselves parallel ; but this is an obvious deduction from J. 28. 

On the assumption that DF, EF will meet Todhunter has this note : " It 
has been proposed to show this in the following way; join DE\ then the 
angles EDFi.nA. DEFax^ together less than the angles ADFand AEF, that 
is, they are together less than two right angles ; and therefore DF and Efi 
will meet, by Axiom 1 2 [Post. 5]. This assumes that ADE and AED are 
acute angles ; it may, however, be easily shown that DE is parallel to BC, so 
that the triangle ADE is equiangular to the triangle ABC; and we must 
therefore select the two sides AB and A C such that ABC and ACB may he 
acute angles." 

This is, however, unsatisfactory, Euchd makes no such selection in tti. 9 
and III. 10, where the same assumption is tacitly made; and it is unnecessary, 
because it is easy to prove that the straight lines DF, EF meet in all cases, 
by considering the different possibilities separately and drawing a separate 
figure for each case. 

Sim son thinks that Euclid's demonstration had been spoiled by some 
unskilful hand both because of the omission to prove that the perpendicular 
bisectors meet, and because "without any reason he divides the proposition 
into three cases, whereas one and the same construction and demonstmtion 
serves for them all, as Cam pan us has observed," However, up to the usual 
words awiji (S(i jrmtjcrat there seems to be no doubt about the text. Heiberg 
suggests that Euclid gave separately the case where /"falls on BC because, in 
that case, only -^Z" needs to be drawn and not BF, CF a.s well. 

The addition, though given in Simson and the text-books as a "corollary," 
has no heading jropio-/t« in the best mss. ; it is an explanation like that which 
is contained in the penultimate paragraph of iii. 25. 

The Greek text has a further addition, which is rejected by Heiiwrg as not 
genuine, "So that, further, when the given angle happens to be less than a 
right angle, DF, EF will fall within the triangle, when it is right, on BC, and, 
when it is greater than a right angle, outside BC. (being) what it was required 
to do." Simson had already observed that the text here is vitiated " where 
mention is made of a given angle, though there neither is, nor can he, any- 
thing in the proposition relating to a given angle." 




IV. s, 6] PROPOSITIONS s, 6 ( 

Proposition 6. 

In a given circle to inscribe a square. • ' 

Let A BCD be the given circle ; 
thus ii is required to inscribe a square in the circle A BCD. 

Let two diameters AC, BD of the 
circle ABCD be drawn at right angles 
to one another, and let AB, BC, CD, 
DA be joined. 

Then, since BE is equal to ED, for 
E is the centre, 

and EA is common and at right angles, 
therefore the base AB is equal to the 
base AD. [[. 4] 

For the same reason 
each of the straight lines BC, CD is also equal to each of 
the straight lines AB, AD ; 

therefore the quadrilateral ABCD is equilateral, 

I say next that it is also right-angled. 

For, since the straight line BD is a diameter of the circle 
ABCD, • 

therefore BAD is a semicircle ; 

therefore the angle BAD is right. [ni. 31] 

For the same reason 
each of the angles ABC, BCD, CD A is also right ; 

therefore the quadrilateral ABCD is right-angled. 

But it was also proved equilateral ; 
therefore it Is a square ; [1. Def. 2»] 

and it has been inscribed in the circle ABCD. 

Therefore in the given circle the square ABCD has been 

inscribed. ■ " 

Q. E. F. 

Euclid here proceeds to consider problems conespondtng to those in 
Props. 2 — s with reference to figures of four or more sides, but with the 
difference that, whereas he dealt with triangles of any fomn, he confines 
him^lf henceforth to regular figures. It happened to be as easy to divide a 
circle into thrct parts which are in the ratio of the angles, or of the supplements 
of the angles, of a triangle as into three c^wn/ parts. But, when it is required to 
inscribe in a circle a figure equiangular to a given quadrUattral, this can only be 



9* 



BOOK IV 



[iv. 6, 7 



donu provided (hat the quadritateral has either pair of opposite angles equal 
to two right angles. Moreover, in this case, tlie problem may be solved in the 
same way as that of iv, z, i.e. by simply inscribing; a triangle equiangular to one 
of the triangles into which the quadrilateral is divided by either diagonal, and 
then drawing on the side corresponding to the diagonal as base another 
triangle equiangular Co the other triangle contained in the quadrilateral. But 
this is not the on/y solution ; there are an infinite 
number of other solutions in which the inscribed 
quadrilateral will, unlike that found by this particular 
method, not he of the same /arm as the given quadri- 
lateral For suppose A BCD to be the quadrilateral 11/ ^•'^ l^lry 

inscribed in the circle by the method of iv. 2. Take '■^'^ '"^ 

any point ff on AB, join AB", and then make the 

angle DAD (measured towards AC) equal to the 

angle BAff. Join ffC, CU. Then AECD is also 

tquiangular to the given quadrilateral, but not of the 

same form. Hence the problem is indeterminate in the case of the general 

quadrilateral. It is equally so if the given quadrilateral is a rectangle ; and it 

is determinate only when the given quadrilateral is a square. 



I \ 



Proposition 7. 

Aboui a given circle to circumscribe a square. 
Let ABCD be the given circle ; 

thus it is required to circumscribe a square about the circle 
ABCD. 

Let two diameters AC, BD of the 
circle ABCD be drawn at right angles 
to one another, and through the points 
A, B, C, D let FG, GH, HK, KF be 
drawn touching the circle ABCD. 

[III. i6,_Por.] 

Then, since FG touches the circk 
ABCD, 

and EA has been joined from the centre 
E to the point of contact at A, 

therefore the angles at A are right. [11 1. 18] 

For the same reason 

the angles at the points B, C, D are also right. 

Now, since the angle AEB is right, 
and the angle EBG is also right, 

therefore GH is parallel to AC. [i. 18^ 




IV. 7] PROPOSITIONS 6, 7 93 

For the same reason 

AC is also parallel to FK, 
so that G// is also parallel to FK. . . ['• 3°] 

Similarly we can prove that 

each of the straight lines GF, HK is parallel to BED, 

Therefore GK, GC, AK, FB, BK are parallelograms ; 
therefore GF is equal to HK, and GH to FK. [i. 34] 

And, since AC is equal to BD, 
and AC is also equal to each of the straight lines G//, FA', 

while BO is equal to each of the straight lines GF, HK, 

[' 34] 
therefore the quadrilateral FGHK is equilateral. 

I say next that it is also right-angled. 

For, since GBEA is a parallelogram, 
and the angle AEB is right, 
therefore the angle AGB is also right. ^ [i. 34] 

Similarly we can prove that 

the angles at H, K, F are also right. 

Therefore FGHK is right-angled. 

But it was also proved equilateral ; 

therefore it is a square ; 
and it has been circumscribed about the circle ABCD. 

Therefore about the given circle a square has been 
circumscribed. 

. ■'^-" Q. E. F. 

I[ is just as easy to describe about a given circle a polygon equiangular to 
any given polygon as it is to describe a square about a given circle. We have 
only to use the method of iv. 3, i.e. to take any radius of the circle, to 
measure round the centre successive angles in one and the same direction 
equal to the supplements of the successive angles of the given polygon and, 
lastly, to draw tangents to the circle at the extremities of the several radii so 
detemiined ; but again the polygon would in general not be of the same form 
as the given one ; it would only be so if the given polygon happened to be 
such that a circle couid be inscribed in it. To take the case of a quadrilateral 
only : it is easy to prove that, if a quadrilateral be described about a circle, 
the sum of one pair of opjxtsite sides must be equal to the sum of the other 
pair. It may be proved, conversely, tliat, if a quadrilateral has the sums of the 
pairs of opposite sides equal, a circle can be inscribed in it. If then a given 
quadrilateial has the sums of the pairs of opposite sides equal, a quadrilateral 
can be described about any given circle not only equiangular with it but 
having the iaxa^form or, in the words of Book vi., similar to it. 



BOOK IV 



[iv. S 




[•• 34] 



Proposition 8. 

In a given square to inscribe a circle. 

Let ABCD be the given square ; 
thus it is required to inscribe a circle in the given square 
ABCD. ^ 

Let the straight lines AD^ AB be 
bisected at the points E, F respectively 

[»• to]. 
through E let EH be drawn parallel 
to either AB or CD, and through 
F let FK be drawn parallel to either 
AD or BC; [1.31] 

therefore each of the figures AK, KB, 
AH, HD, AG, GC, BG, GD is a parallelogram, 
and their opposite sides are evidently equal. 

Now, since AD is equal to AB, 
and ^^ is half of AD, and AF half of AB, 

therefore AE is equal to AF, 
so that the opposite sides are also equal ; • ' - -■ 
therefore FG is equal to GE. 

Similarly we can prove that each of the straight lines GH, 
GK is equal to each of the straight lines FG, GE ; 

therefore the four straight lines GE, GF, GH, GK are 
equal to one another. 

Therefore the circle described with centre G and distance 
one of the straight lines GE, GF, . GH, GK will pass also 
through the remaining points. 

And it will touch the straight lines AB, BC, CD, DA, 
because the angles at E, F, H, K are right. 

For, if the circle cuts AB, BC, CD, DA, the straight 
line drawn at right angles to the diameter of the circle from 
its extremity will fall within the circle : which was proved 
absurd ; [iii, ifi) 

therefore the circle described with centre G and distance 
one of the straight lines GE, GF, GH, GK will not cut 
the straight lines AB, BC, CD, DA. 

Therefore it will touch them, and will have been inscribed 
in the square ABCD. 

Therefore in the given square a circle has been inscribed. 




IV. 8, 9] PROPOSITIONS 8, 9 9$ 

As was remarked in the Iftst note, a circle can be inscribed in any 
^uadrilatiral -vihKh has the sum of one pair ofoppc^ite sides equal to the sum 
of the other pair. In particular, il follows that a circle can be inscribed in a 
tfuare or a rhsmbus^ but not in a rectangle or a rhomboid. 



' * Proposition 9, 

About a given square to circumscribe a circle. 
Let A BCD be the given square ; 

thus it is required to ctrcu inscribe a circle about the square 

A BCD. 

For let AC, BD be joined, and let them 
cut one another at E, 

Then, since DA is equal to AB^ 
and AC is common, 

therefore the two sides DA, AC are equal 
to the two sides BA, AC; 
and the base DC is equal to the base BC ; 
therefore the angle DAC is equal to 
the angle BAC. [i. 8] 

Therefore the angle DAB is bisected by AC. 

Similarly we can prove that each of the angles ABC, 
BCD, CDA is bisected by the straight lines AC, DB. 

Now, since the angle DAB is equal to the angle ABC, 
and the angle BAB is half the angle DAB, 
and the angle BBA half the angle ABC, 'V 

therefore the angle BAB is also equal to the angle £BA ; 
so that the side £A is also equal to BB. [i. 6] 

Similarly we can prove that each of the straight lines 
BA, BB is equal to each of the straight lines EC, ED. 

Therefore the four straight lines EA, BB, BC, BD are 
equal to one another. 

Therefore the circle described with centre E and distance 
one of the straight lines EA, EB, EC, ED will pass also 
through the remaining points ; 
and it will have been circumscribed about the square ABCD. 

Let it be circumscribed, as ABCD. 

Therefore about the given square a circle has been 
circumscribed. 




96 BOOK IV [iv. lo 

PROPOSITtON 10. '■'- . ■ .• 

Ta cottslrticl an isosceles triangle Itaving each of the angles 
at the base double of the remaining one. 

Let any straight line AB be set out, and let it be cut at 
the point C so that the rectangle 
contained by AB, BC is equal to 
the square on CA\ [•'■"] 

with centre A and distance AB let 
the circle BDE be described, 

and let there be fitted in the circle 
BDE the straight line BD equal to 
the straight line AC which is not 
greater than the diameter of the 
circle BDE, [iv. ,] 

Let AD, DC be joined, and let 
the circle ACD be circumscribed about the triangle A CD. 

['V. S) 
Then, since the rectangle AB, BC is equal to the square 
on AC, 

and AC is equal to BD, 

therefore the rectangle AB, BC is equal to the square on BD. 

And, since a point B has been taken outside the circle 
ACD, 

and from B the two straight lines BA, BD have fallen on 
the circle ACD, and one of them cuts it, while the other falls 
on it, 

and the rectangle AB, BC is equal to the square on BD, 

therefore BD touches the circle ACD. [in. 37] 

Since, then, BD touches it, and DC is drawn across 
from the point of contact at D, 

therefore the angle BDC is equal to the angle DAC in the 
alternate segment of the circle. [in. 3*] 

Since, then, the angle BDC is equal to the angle DAC, 
let the angle CDA be added to each ; 

therefore the whole angle BDA is equal to the two angles 
CDA, DAC. 



IV, to] PROPOSITION ro 



97 



But the exterior angle BCD is equal to the angles CD A, 
DAC; [1.3a] 

therefore the angle BDA is also equal to the angle BCD. 

But the angle BDA is equal to the angle CBD. since the 
side AD is also equal to AB ; [i. s] 

so that the angle DBA is also equal to the angle BCD. 

Therefore the three angles BDA, DBA, BCD are equal 
to one another. 

And, since the angle DBC is equal to the angle BCD, 

the side BD is also equal to the side DC. [i. 6] 

But BD is by hypothesis equal to CA ; 
therefore CA is also equal to CD, 

so that the angle CD A is also equal to the angle DA C ; 

[i-S] 
therefore the angles CD A, DAC are double of the angle DA C. 

But the angle BCD is equal to the angles CD A, DAC; 

therefore the angle BCD is also double of the angle CAD. 

But the angle BCD is equal to each of the angles BDA, 
DBA , 

therefore each of the angles BDA, DBA is also double of 
the angle DAB. 

Therefore the isosceles triangle ABD has been constructed 
having each of the angles at the base DB double of the 
remaining one. 

Q. E. F. 

There is every reason to conclude that the connexion of the triangle 
constructed in this proposition with the regular pentagon, and the construction 
of the triangle itself, were the discovery of the Pythagoreans. In the first 
place the Scholium iv. No. a {Heiberg, Vol. v. p. 273) says " this Book is the 
discovery of the Pythagoreans." Secondly, the summary in Proclus (p. fi$, to) 
says that Pythagoras discovered "the construction of the cosmic figures," by 
which rnust be understood the five regular solids. Thirdly, lamblichus {yn. 
Pyth, c. 18, s. 38) quotes a story of Hippasus, " that he was one of the Pytha- 
goreans but, owing to his being the first to publish and write down (the con- 
struction of) the sphere arising from the twelve pentagons {rrpi in tuv hi&txa 
ir<kraywfijf), perished by shipwreck for his inipiety, having got credit for the 
discovery all the same, whereas everything belonged to HIM {intivcmrov dkSpot), 
for it is thus that they refer to Pythagoras, and they do not call him by his 
name." Cantor has (i,, pp. 176 sqq.) collected notices which help us to form 
an idea how the discovery of the Euclidean construction for a regular pentagon 
may have been arrived at by the Pythagoreans. 

Plato puts into th". mouth of Timaeus a description of the formation from 

\ 



BOOK IV 



[iv. 10 




right-angled triangles of the figures which are the faces o( the first four regular 

solids. The face of the cube is the S(]uare which is formed from isosceles 

right-angled triangles by placing four of these triangles contiguously so that 

the four right angles are in contact at the centre. The 

equilateral triangle, however, which is the form of the faces of 

the tetrahedron, the octahedron and the icosahedron, cannot 

be constructed from isosceles right-angled triangles, but is 

constructed from a particular scalene right-angled triangle 

which Timaeus (54 a, h) regards as the most b<:autiful of all 

scalene right-angled triangles, namely that in which the square on one of the 

sides about the right angle is three times the square on the other. This is, of 

course, the triangle forming half of an equilateral triangle bisected by the 

perpendicular from one angular point on the opposite side. The Platonic 

Timaeus does not construct his equilateral triangle from two such triangles 

but from six, by placing th« latter contiguously round a 

point so that the hypotenuses and the smaller of the sides 

about the right angles respectively adjoin, and all of them 

meet at the common centre, as shown in the figure 

{T/macHs, 54 d, k.). The probability that this exposition 

was Pythagorean is confirmed by the independent testimony 

of Proclus (pp. 304—5), who attributes to the Pythagoreans 

the theorem that six equilateral triangles, or three hexagons, or four squar^ 

placed contiguously with one angular point of each at a common point, will 

just fill up the four right angles round that point, and that no other regular 

polygons in any numbers have this property. 

How then would it be proposed to split up into triangles, or to make up 
out of triangles, the face of the remaining solid, the dodecahedron ? It would 
easily be seen that the pentagon could not be constructed by means of tlie 
two right-angled triangles which were used for constructing the square and the 
equilateral triangle respectively. But attempts would naturally be made to 
split up the pentagon into elementary triangles, and traces of such attempts 
are actually forthcoming. Plutarch has in two passages spoken of the division 
of the faces of the dodecahedron into triangles, remarking in one place 
(Quaes/. Platon. v. i) that each of the twelve faces is made up of 30 elemen- 






tary scalene triangles, so that, taken together, they give 360 such triangles, 
and in another {Dt deftctu oramlorum, c. 33) that the elementary triangle of 
the dodecahedron must be different from that of the tetrahedron, octahedron 
and icosahedron. Another writer of the and cent,, Alcinous, has, in his 
introduction to the study of Plato {De doctrina Platonis, c. 11), spoken 
similarly of the 360 elements which are produced when every one of the 
pentagons is divided into 5 isosceles triangles, and each of the latter into 
6 scalene triangles. Now, if we proceed to draw lines in a pentagon sejmrating 
it into this number of small triangles as shown in the above figure, the figure 



IV. lo] PROPOSITION lo 99 

which stands out most prominently in the mass of lines is the "star-pentagon," 
as drawn separately, which then (if the consecutive comers be joined) suggests 
the drawing, as part of a pentagon, of a triangle of a definite character. Now 
we are expressly told by Lucian and the scholiast to the Clouds of Aristophanes 
(see Bretschneider, pp. 85 — 86) that the triple interwoven triangle, the penta- 
gram (to TpurXflEf Tpiycuvov, TO Zi oAAijAaic, to TroTaypttfiftov), was used by the 
Pythagoreans as a symbol of recognition between the members of the same 
school {av^ohjjf ipos Tous ofioSo'fou! ixf™"'°)i *"d '"'** called by them Health. 
There seems lo be therefore no room for doubt that the construction of a 
pentagon by means of an isosceles triangle having each of its base angles 
double of the vertical angle was due to the Pythagoreans. 

The construction of this triangle depends upon ii. 1 1, or the problem of 
dividing a straight tine so that the rectangle contained by the whole and one 
of the parts is equal to the square on the other part. This problem of course 
appears again in Eucl. vi. 30 as the problem of cutting a given straight line in 
exirtme and mean ratio, i.e. the problem of the goldtn section, which is nc- 
doubt " the section " referred to in the passage of the summary given by 
Proclus (p. 67, 6) which says that Eudoxus "greatly added to the number 
of the theorems which Plato originated regarding the section." This idea that 
Plato began the study of the " golden section " as a subject in itself is not in 
the least inconsistent with the supposition that the problem of Eucl ii. 1 1 was 
solved by the Pythagoreans. The very fact that Euclid places it among other 
propositions which are clearly Pythagorean in origin is significan|, as is also 
the fact that its solution is effected by " applying to a straight line a rectangle 
equal to a given square and exceeding by a square," while Proclus says plainly 
(p. 419, 15) that, according to Eudemus, "the application of areas, their 
txcuding and their falling short, are ancient and discoveries of the Muse of 
the Pythagoreans." 

We may suppose the construction of iv. to to have been arrived at by 
analysis somewhat as follows (Todhunter's Euclid, p. 325). 

Suppose the problem solved, i.e. let ABD be an isosceles triangle having 
each of its base angles double of the vertical angle. 

Bisect the angle ADB by the straight line DC meeting AB in C. [1, 9] 

Therefore the angle BDC is equal to the angle BAD ; and the angle 
CDA is also equal to the angle BAD, 

so that DC is equal to CA, 

Again, since, in the triangles BCD, BDA, 

the angle BDC is equal to the angle BAD, 
and the angle B is common, 

the third angle BCD is equal to the third angle BDA, and therefore to 
the angle DEC. 

Therefore Z'C is equal to DB. 

Now, if a circle be described about the triangle ACD [iv. 5}, since the 
angle BDC is equal to the angle in the segment CAD, 

BD must touch the circle [by the converse of [ii. 32 easily proved from it 
by riduttio ad aiiurdum\ 

Hence [11 [. 36] the square oa BD and therefore the square on CD, or 
AC, is equal to the rectangle AS, BC. 

Thus the problem is reduced to that of cutting AB at C so that the 
rectangle AB, BC is equal to the square on AC, [ii, 11] 



t99 ' BOOK IV [iv. lo, It 

When this is done, we have only to draw a circle with centre A and radius 
A£ and place in it a chord SU equal in length to AC. [iv. i] 

Since each of the angles ABD, ADB is double of the angle BAD, the 
latter is equal to one-fifth of the sum of all three, i.e. is one-fifth of two right 
angles, or two-fifths of a right angle, and each of the base angles is four-fifths 
of a right angle. 

If we bisect the angle BAD, we obtain an angle equal to one-fifth of a 
right angle, so that the proposition enables us to divide a right angle into five 
equal parts. 

It will be observed that BD is the side of a regular dtcagen inscribed in 
the larger circle. 

Proclus, as retnarked above (Vol. i, p. 130), gives iv. 10 as an instance in 
which two of the six formal divisions of a proposition, the idting-out and the 
"definition" are left out, and explains that they are unnecessary because 
there is no datum in the enunciation. This is however no more than formally 
true, because Euclid does begin bis proposition by tetling out " any straight 
line AB" and he constructs an isosceles triangle having AB for one of its 
equal sides, i.e. he does practically imply a datum in the enunciation, and a 
corresponding setting-out and "definition '' in the proposition itself. 



Proposition 11. 

In a given circle to inscribe an equilateral and ^uiangular 
pentagon. 

Let ABCDE be the given circle ; 
thus it is required to inscribe in the circle ABCDE an equi- 
lateral and equiangular pentagon. 

Let the isosceles triangle FGH 
be set out having each of the angles 
at G, H double of the angle at F\ 

[iv. 10] 

let there be inscribed in the circle 
ABCDE the triangle ACD equi- 
angular with the triangle FGH, so 

that the angle CAD is equal to the angle at F and the angles 
at G, ^respectively equal to the angles ACD, CD A ; [iv. a] 
therefore each of the angles ACD, CD A is also double of the 
angle CAD. 

Now let the angles ACD, CD A be bisected respectively 
by the straight lines CE, DB [1. 9], and let AB, BC, DE, EA 
be joined. 

Then, since each of the angles ACD, CD A is double of 
the angle CAD, 
and they have been bisected by the straight lines CE, DB, 





IV. ii] PROPOSmONS 10, II i«i 

therefore the five angles DAC, ACE, BCD, CDS, BDA 
are equal to one another. 

But equal angles stand on equal circumferences ; [iii. a6] 

therefore the five circumferences AB, BC, CD, DE, EA are 
equal to one another. 

But equal circumferences are subtended by equal straight 
lines ; [m. 19] 

therefore the five straight lines AB, BC, CD, DE, EA are 
equal to one another ; 

therefore the pentagon ABCDE is equilateral. 

I say next that it is also equiangular. 
For, since the circumference AB is equal to the circum- 
ference DE, let BCD be added to each ; 

therefore the whole circumference ABCD is equal to the 
whole circumference EDCB. 

And the angle A ED stands on the circumference ABCD, 
and the angle BAE on the circumference EDCB ; 

therefore the angle BAE is also equal to the angle AED. 

[iiL a;] 
For the same reason * 

each of the angles ABC, BCD, CDE is also equal to each 
of the angles BAE, AED ; 

therefore the pentagon ABCDE is equiangular. 

But it was also proved equilateral ; 

therefore in the given circle an equilateral and equi- 
angular pentagon has been inscribed. 

Q. E. F. 



De Morgan remarks that " the method of iv. 11 is not %a natuial as 
malting a direct use of the angle obtained in the last." On the other hand, 
if we look at the figure and notice that it shows the whole of the pmtagrafo- 
star except one line {that connecting B and E), I think we shall conclude 
that the method is nearer to that used by the Pythagoreans, and therefore of 
much more historical interest. 

Another method would of course be to use iv. 10 to describe a decagtnt in 
the circle, and then to join any vertex to the next alternate one, the tatter to 
the next alternate one, and so on. 



K». 



BOOK IV 



[iV. II, I » 



Mr H. M. Taylor gives "a complete geometrical construction for in- 
scribing a regular decagon or pentagon in a given circle," as follows. 

" Find O the centre. 

Draw two diameters AOC, BOD at right g 

angles to one another. 

Bisect OD in £. 

Draw A£ and cut off E£ equal to 0£. 

Place round the circle ten chords equal 
to AF. 

These chords will be the sides of a regular 
decagon. Draw the chords joining Hve alternate 
vertices of the decagon ; they will be the sides 
of a regular pentagon." 

The construction is of course only a com- 
bination of those in ]i. ii and iv. i ; and the 
proof would have to follow that in iv. lo. 




Proposition 12, 

About a given circle to circumscribe an equilateral and 
equiangular pentagon. 

Let y4 .5 CZ?.£' be the given circle ; >- 

thus it is required to circumscribe an equilateral and equi- 
angular pentagon about the circle 
ABCDE. 

Let A, B, C, D, E be conceived to 
be the angular points of the inscribed 
pentagon, so that the circumferences 
AB, EC, CD, DE, EA are equal ; 

through A, B, C, D. E let G//, HK, 

KL, LM, MG be drawn touching the 

circle ; [in. 16, Por.] 

let the centre F of the circle ABCDE be taken [m. 1], and 

let FB, FK, FC, FL, FD be joined. 

Then,since the straight line KL touches the circle ABCDE 
at C, 

and FC has been joined from the centre F to the point of 
contact at C, 

therefore FC is perpendicular to KL ; , [in. 18] 

therefore each of the angles at C is right. • ■ 

For the same reason 

the angles at the points B, D are also right 




IV. laj PROPOSITIONS ii, la 103 

And, since the angle FCK is right, 
therefore the square on FK is equal to the squares on FC, CK. 

For the same reason [1. 47] 

the square on FK is also equal to the squares on FB, BK ; 

so that the squares on FC, CK are equal to the squares 
on FB, BK, 

of which the square on FC is equal to the square on FB ; 
therefore the square on CK which remains is equal to the 
square on BK. 

Therefore BK is equal to CK. 

And, since ^5 is equal to T^C • • 

and FK common, 

the two sides BF, FK are equal to the two sides CF, FK \ 
and the base BK equal to the base CK ; 

therefore the angle BFK is equal to the angle KFC, [i. 8] 

and the angle BKF to the angle FKC. 
Therefore the angle BFC is double of the angle KFC, 

and the angle BKC of the angle FKC. 

For the same reason 

the angle CFD is also double of the angle CFL, 

and the angle DLC of the angle FL C. 

Now, since the circumference BC is equal to CD, 
the angle BFC is also equal to the angle CFD. [in. 17] 

And the angle BFC is double of the angle KFC, and the 
angle DFC of the angle LFC ; 

therefore the angle KFC is also equal to the angle LFC. 

But the angle FCK is also equal to the angle FCL ; 
therefore FKC, FLC are two triangles having two angles 
equal to two angles and one side equal to one side, namely 
FC which is common to them ; 

therefore they will also have the remaining sides equal to the 
remaining sides, and the remaining angle to the remaining 
angle ; [i- *6) 

therefore the straight line KC is equal to CL, 
and the angle FKC to the angle FLC, 

And, since KC is equal tO CZ, , 

therefore KL is double of KC , . 



104 BOOK IV [iv. ti, 13 

For the same reason it can be proved that 

I/J^ is also double of S/C. 1 

And ^A' is equal to ^C; ■ 

therefore I/K is also equal to KL. 

Similarly each of the straight lines //G, GM, ML can 
also be proved equal to each of the straight lines //K, KL ; 

therefore the pentagon GHKLM is equilateral. 
. - I say next that it is also equiangular. 

For, since the angle FKC is equal to the angle FLC, 
and the angle HKL was proved double of the angle FKC, 

and the angle KLM double of the angle FLC, 
therefore the angle HKL is also equal to the angle KLM. 

Similarly each of the angles KHG, HGM, GML can also 
be proved equal to each of the angles HKL, KLM; 
therefore the five angles GHK, HKL, KLM, LMG, MGH 
are equal to one another. 

Therefore the pentagon GHKLM is equiangular. 

And it was also proved equilateral ; and it has been 
circumscribed about the circle ABCDE. 

Q. E. F. 

De Morgan remarks that iv. 12, 13, 14 supply the pkce of the following : 
Having given a regular polygon of any number of sides inscribed in a circle, lo 
describe the same about ike circle; and, having given the polygon, lo inscribe and 
circumscribe a circle. For the method can be applied generally, as indeed 
Euclid practically says in the Porism to iv. 15 about the regular hexagon and 
in the remark appended to iv. 16 about the regular fifteen-angled figure. 

The conclusion of this proposition, " therefore about the given circle an 
equilateral and equiangular pentagon bas been circumscnbed," is omitted in 
the Mss. 



Proposition 13. 

In a given pentagon, wkick is equilateral and equiangular, 
to inscribe a circle. 

Let ABCDE be the given equilateral and equiangular 
pentagon ; 

thus it is required to inscribe a circle in the pentagon 
ABCDE. 

For let the angles BCD, CDE be bisected by the 
straight lines CF, Z?J^ respectively ; and from the point F, at 



IV. 13] 



PROPOSITIONS 12, 13 



105 




■'IJ-/J 



which the straight lines CF, DF meet one another, let the 
straight lines FB, FA, FE be joined. 1 • • 

Then, since BC is equal to CD, 
and CV^ common, 

the two sides BC, CF are equal to the 
two sides DC, CF\ 

and the angle BCF is equal to the 
angle DCF; 

therefore the base BF is equal 
to the base DF, 

and the triangle BCF is equal to the 
triangle DCF, 

and the remaining angles will be equal to the remaining angles, 
namely those which tne equal sides subtend. [1. 4] 

Therefore the angle CBF is equal to the angle CDF. 

And, since the angle CDE is double of the angle CDF, 
and the angle CDE is equal to the angle ABC, 
while the angle CDF is equal to the angle CBF\ 
therefore the angle CBA is also double of the angle CBF) 
therefore the angle ABF is equal to the angle FBC ; 
therefore the angle ABC has been bisected by the straight 
line BF. 

Similarly it can be proved that 
the angles BAE, AED have also been bisected by the straight 
lines FA, FE respectively. 

Now let EG, FH, FK, FL, FMh& drawn from the point 
F perpendicular to the straight lines AB, BC, CD, DE, EA. 

Then, since the angle HCF is equal to the angle KCF, 
and the right angle FHC is also equal to the angle FKC, 
FHC, FKC are two triangles having two angles equal to two 
angles and one side equal to one side, namely EC which is 
common to them and subtends one of the equal angles ; 
therefore they will also have the remaining sides equal to the 
remaining sides ; [i. i6] 

therefore the perpendicular FH is equal to the perpendicular 
FK. 

Similarly it can be proved that 
each of the straight lines FL, FM, EG is also equal to each 
of the straight lines EH, FK ; 



i«6 BOOK IV [iv. 13, u 

therefore the five straight lines FG, FH, FK, FL, FM are 
equal to one another. 

Therefore the circle described with centre F and distance 
one of the straight lines FG, FH, FK, FL, FM will pass 
also through the remaining points ; 

and it will touch the straight lines AB, BC, CD, DE, EA, 
because the angles at the points G, N, K, L, M aire right. 

For, if it does not touch them, but cuts them, 

it will result that the straight line drawn at right angles to 
the diameter of the circle from its extremity falls within the 
circle : which was proved absurd. [ui. 16] 

Therefore the circle described with centre F and distance 
one of the straight lines FG, FN, FK, FL, FM will not 
cut the straight lines AB, BC, CD, DE, EA ; 

therefore it will touch them. 

Let it be described, as GHKLM. 

Therefore in the given pentagon, which is equilateral and 
equiangular, a circle has been inscribed, 

Q, E. F. 



Proposition 14. 

About a given pentagon, which ts equilateral and equi- 
angular, to circumscribe a circle. 

Let ABCDE be the given pentagon, which is equilateral 
and equiangular ; 

thus it is required to circumscribe a circle 
about the pentagon ABCDE. 

Let the angles BCD, CDE be bisected 
by the straight lines CF, OF respectively, 
and from the point F, at which the straight 
lines meet, let the straight lines FB, FA, 
FE be joined to the points B, A, E. 

Then in manner similar to the pre- 
ceding it can be proved that the angles 
CBA, BAE, AED have also been bisected by the straight 
lines FB, FA, FE respectively. • 




IV. 14, is] 



PROPOSITIONS 13— IS 



107 



Now, since the angle BCD is equal to the angle CDE, 
and the angle FCD is half of the angle BCD, 
and the angle CDF half of the angle CDE, 
therefore the angle FCD is also equal to the angle CDF, ■' ! 

so that the side FC is also equal to the side FD. [i- 6] 

Similarly it can be proved that 
each of the straight lines FB, FA, FE is also equal to each 
of the straight lines FC, FD ; 

therefore the five straight lines FA, FB, FC, FD, FE are 
equal to one another. 

Therefore the circle described with centre F and distance 
one of the straight lines FA, FB, FC, FD, FE will pass 
also through the remaining points, and will have been 
circumscribed. , , 

Let it be circumscribed, and let it be ABCDE. 

Therefore about the given pentagon, which is equilateral 
and equiangular, a circle has been circumscribed. 

Q. E, F. 



Proposition 15. 

In a given, circle to inscribe an equilateral and equiangular 
hexagon. 

Let ABCDEF be the given circle ; ,...::; 

thus it is required to inscribe an equilateral and equiangular 
hexagon in the circle ABCDEF. 

Let the diameter AD of the circle 
ABCDEF be drawn ; 
let the centre G of the circle be taken, and 
with centre D and distance DG let the 
circle EGCH be described ; 
let EG, CG be joined and carried through 
to the points B, F, 

and let AB, EC, CD, DE, EF, FA be 
joined. 

I say that the hexagon ABCDEF is 
equilateral and equiangular. 

For, since the point G is the centre of the circle ABCDEF, 
GE is equal to GD. ■■ v. •• v 




•»*S BOOK IV 1 " [iv, IS 

Again, since the point D is the centre of the circle GCH, 

DE is equal to DG, ^ ■':-' 

„ But GE was proved equal to GD ; 

therefore GE is also equal to ED ; 

therefore the triangle EGD is equilateral ; 

and therefore its three angles EGD, GDE, DEG are equal 
to one another, inasmuch as, in isosceles triangles, the angles 
at the base are equal to one another. [i. s] 

And the three angles of the triangle are equal to two 
right angles ; [i. 3^1 

therefore the angle EGD is one-third of two right angles. 

Similarly, the angle DGC can also be proved to be one- 
third of two right angles. 

And, since the straight line CG standing on EB makes 
the adjacent angles EGC, CGB equal to two right angles, 

therefore the remaining angle CGB is also one-third of two 
right angles. 

Therefore the angles EGD, DGC, CGB are equal to one 
another ; 

so that the angles vertical to them, the angles SGA, AGF, 
FGE are equal, [i. 15] 

Therefore the six angles EGD, DGC, CGB, BGA, AGF, 
FGE are equal to one another. 

But equal angles stand on equal circumferences ; {m- a6] 
therefore the six circumferences AB, BC, CD, DE, EF, FA 
are equal to one another. 

And equal circumferences are subtended by equal straight 
lines ; [m. 29] 

therefore the six straight lines are equal to one another; 

therefore the hexagon ABCDEF is equilateral, 

I say next that it is also equiangular. 

For, since the circumference FA is equal to the circum- 
ference ED, 

let the circumference ABCD be added to each ; 

therefore the whole FA BCD is equal to the whole 
EDCBA ; 



IV, is] proposition is 109 

and the angle FED stands on the circumference FA BCD, 

and the angle AFE on the circumference EDCBA ; 

therefore the angle AFE is equal to the angle DEF, 

[m. 27] 

Similarly it can be proved that the remaining angles of 
the hexagon ABCDEF are also severally equal to each of 
the angles AFE, FED ; 

therefore the hexagon ABCDEF is equiangular. 

But it was also proved equilateral ; 

and it has been inscribed in the circle ABCDEF. 

Therefore in the given circle an equilateral and equiangular 
hexagon has been inscribed. 

Q. E. F. 

PoRiSH. From this it is manifest that the side of the 
hexagon is equal to the radius of the circle. 

And, in like manner as in the case of the pentagon, if 
through the points of division on the circle we draw 
tangents to the circle, there will be circumscribed about the 
circle an equilateral and equiangular hexagon in conformity 
with what was explained in the case of the pentagon. 

And further by means similar to those explained in the 
case of the pentagon we can both inscribe a circle in a given 
hexagon and circumscribe one about it, , 

Q. E. F. 

Hetberg, I think with good reason, considers the Porism to this proposition 
to be referred to in the instance which Proclus (p. 304, a) gives of a porism 
following a problem. As the text of Proclus stands, " the (poristn) found 
in the second Book (td ii ir ry Smrifnf ^ijSXi'u xtiiititor) is a porism to a 
problem " ; but this is not true of the only porism that we find in the second 
Book, namely the jwrism to it. 4. Hence Heibeig thinks that for rif 
StvfifMf fiifiXiif should be read 1^ £' fii^Kuf, i.e. the fourth Book. Moreover 
Proclus speaks of tAe porism in the particular Book, from which we gather 
that there was only arte porism in BooJt iv. as he knew it, and therefore that 
he did not regard as a porism the addition to iv. 5. Cf. note on that 
proposition. 

It appears that Theon substituted for the first words of the Porism to 
IV. 15 "And in like manner as in the case of the pentagon" (d/iouiit Si 
ToTs M. rol leivTaymrm} the simple word " and " or " also " (Wj, apparently 
thinking that the words had the same meaning as the similar words lower 
down. This is however not the case, the meaning being that " if, as in the 
case of the pentagon, we draw tangents, we can prove, also as was done in 
the case of the pentagon, that the figure so formed is a circumscribed r^ular 
hexagon." 



BOOK. IV 
Proposition i6. 



[tv. i6 




In a given circle to inscribe a fifteen-angled figure which 
shall be both equilateral and equiangular. 

Let ABCD be the given circle ; 
thus it is required to inscribe in the circle ABCD a fifteen- 
angled figure which shall be 
both equilateral and equi- 
angular. 

In the circle ABCD let 
there be inscribed a side AC 
of the equilateral triangle 
inscribed in it, and a side AB 
of an equilateral pentagon ; 
therefore, of the equal seg- 
ments of which there are 
fifteen in the circle ABCD, 
there will be five in the cir- 
cumference ABC which is 
one-third of the circle, and 
there will be three in the cir- 
cumference AB which is one-fifth of the circle ; 

therefore in the remainder BC there will be two of the 
equal segments. 

Let BC be bisected at E ; [m. 30] 

therefore each of the circumferences BE, EC is a fifteenth 
of the circle ABCD, 

If therefore we join BE, EC and fit into the circle ABCD 
straight lines equal to them and in contiguity, a fifteen-angled 
figure which is both equilateral and equiangular will have been 
inscribed in it. 

' ' ■ ' Q. E. F. 

And, in like manner as in the case of the pentagon, if 
through the points of division on the circle we draw 
tangents to the circle, there will be circumscribed about the 
circle a fifteen-angled figure which is equilateral and equi- 
angular. 

And further, by proofs similar to those in the case of the 
pentagon, we can both inscribe a circle in the given fifteen- 
angled figure and circumscribe one about it. 



JV. .6] PROPOSITION tfi ui 

Here, as in ii[. lo, we have the term "circle" used by Euclid in its 
exceptional sense of the drcumjerena of a circle, instead of the "plane figurt 
contained by one hne" of i. l)ef. 15. Cf. the note on that definition (Vol. i. 
pp. 184—5}. 

Proclus {p. 269) refers to this proposition in illustratiotv of his statement 
that Euclid gave proofs of a number of propositions with an eye to their use 
in astronomy. " With regard to the last proposition in the fourth Book in 
which he inscribes the side of the fifteen-angled figure in a circle, for what 
object does anyone assert that he propounds it except for the reference of this 
problem to astronomy ? For, when we have inscribed the fifteen -angled figure 
in the circle through the poles, we have the distance from the poles both of 
the equator and the zodiac, since they are distant from one another by the 
side of the fifteen-angled figure," This agrees with what we know from other 
sources, namely that up to the time of Eratosthenes {circa 2iJ4 -204 B.C.) 24 
was generally accepted as the correct measurement of the obliquity of the 
ecliptic. This measurement, and the construction of the fifteen-angled figure, 
were probably due to the Pythagoreans, though it would appear that the 
former was not known to Oenopides of Chios {fl. circa 460 B.C.), as we learn 
from Theon of Smyrna {pp. 198 — 9, ed, Hiller), who gives Dercy Hides as his 
authority, that Eudemus (H. circa 32a B.C.) stated in his dcrTp«A.<ry('<u that, 
while Oenopides discovered certain things, and Thales, Anaximander and 
Anaximenes others, it was the rest (01 AoHrm) who added other discoveries 
to these and, among them, that " the axes of the fixed stars and of the planets 
respectively are distant from one another by the side of a fifteen-angled figure." 
Eratosthen« evaluated the angle to J^rds of 180°, i.e. about 23' 51' 10", 
which measurement was apparently not improved upon in antiquity (cf. Ptolemy, 
Syataxii, ed. Heiberg, p. 68). 

Euclid has now shown how to describe regular polygons with 3, 4, 5, £ 
and 15 sides. Now, when any regular polygon is given, we can construct a 
regular polygon with twice the number of sides by first describing a circle 
about the given polygon and then bisecting all the smaller arcs subtended by 
the sides. Applying this process any number of times, we see that we can by 
Euclid's methods construct regular polygons with 3,1", 4-a*, 5,2", 15.2" sides, 
where « is zero or any positive integer. 



.1 1, , 



\ . f ' ' 



BOOK V. 



INTRODUCTORY NOTE. 

The anonymous author of a scholium to Book v. (Euclid, ed. Heiberg, 
Vol. V. p. 280), who is perhaps E^oclus, tells us that "some say" thb Book, 
containing the general theory of proportion which ts equally applicable to 
geometry, arithmetic, music, and all mathematical science, "is the discovery 
of Eudoxus, the teacher of Plata" Not that there had been no theory of 
proportion developed before his time j on the contrary, it is certain that the 
Pythagoreans had worked out such a theory with regard to numbtrs, by which 
must be understood commensurable and even whole numbers {a number 
being a " multitude made up of units," as defined in Eucl. vii). Thus we 
are told that the Pythagoreans distinguished three sorts of means, the 
arithmetic, the geometric and the harmonic mean, the geometric mean 
being called proportion (amXoyui) par exallenee; and further lamblichus 
speaks of the "most perfect proportion consisting of four terms and specially 
called harmonU" in other words, the proportion 

a + b xab , 

■'■'■■ ■ ■ "'^'IT-y^' 

which was said to be a discovery of the Babylonians and to have been Rrst 
introduced into Greece by Pythagoras (lamblichus, Comm. en Ni&mtachas, 
p. ti8). Now the principle of similitude is one which is presupposed by all 
the arts of design from their very beginnings ; it was certainly known to the 
Egyptians, and it must certainly have been thoroughly familiar to Pythagoras 
and his school. This consideration, together with the evidence of the 
employment by him of the g^emetric proportion, makes it indubitable that the 
Pythagoreans used the theory of proportion, in the form in which it was 
known to them, i.e. as applicable to commensurables only, in their geometry. 
But the discovery, also by the Pythagoreans, of the incommensurable would 
of course be seen to render the proofs which depended on the theory of 
proportion as then understood inconclusive ; as Tannery observes (Xrr 
Giomftrie grecqui, p. 98), *' the discovery of incommensurability must have 
caused a veritable logical scandal in geometry and, in order to avoid it, they 
were obliged to restrict as far as possible the use of the principle of similitude, 
pending the discovery of a means of establishing it on the basis of a theory of 
proportion independent of commensurability." The glory 0/ the latter dis- 
covery belongs then most probably to Eudoxus. Certain it is that the com. 
plete theory was already familiar to Aristotle, as we shall see later. 



V. DEFF. I, a] INTRODUCTORY NOTE uj. 

It seems probable, as indicated by Tannery {lot. a'i.}, that the theory 
of proportions and the principle of similitude took, in the earliest Greek 
geometry, an earlier place than they do in Euclid, but that, in consequence 
of the discovery of the incommensurable, the treatment of the subject was 
fundamentally remodelled in the period between Pythagoras and Eudoxus, 
An indication of this is afforded by the clever device used in Euclid i. 44 
for applying to a given straight line a parallelogram equal to a given triangle ; 
the equality of the "complements" in a parallelc^iam is there used for doing 
what is practically finding a fourth proportional to three given straight lines. 
Thus Euclid was no doubt following for the subject-matter of Books t. — iv. 
what had become the traditional method, and this is probably one of the 
reasons why proportions and similitude are postponed till as late as Books 
v., VI, 

It is a remarkable fact that the theory of proportions is twice treated in 
Euclid, in Book v. with reference to magnitudes in general, and in Book vci. 
with reference to the particular case of numbers. The latter exposition 
referring only to commensurable^ may be taken to represent fairly the theory 
of proportions at the stage which it had reached before the great extension of 
it made by Eudoxus. The differences between the definitions etc. in Books v. 
and VII. will appear as we go on ; but the question naturally arises, why did 
Euclid not save himself so much repetition and treat numbers merely as a 
particular case of magnitude, referring back to the corresponding more 
general propositions of Book v. instead of proving the same propositions 
over again for numbers? It could not have escaped him that numbers 
fall under the conception of magnitude. Aristotle had plainly indicated 
that magnitudes may be numbers when he observed {Anal. post. t. 7, 
75 b 4) that you cannot adapt the arithmetical method of proof to the 
properties of magnitudes if the magnitudes are not numbers. Further 
Aristotle had remarked {Anal. post. 1. 5, 74 a 17) that the proposition that 
the terms of a proportion can be taken alternately was at one time proved 
•eparately for numbers, lines, solids and times, though it was possible to prove 
it for all by one demonstration ; but, because there was no common tuime 
comprehending them all, namely numbers, lengths, times and solids, and their 
character was different, they were taken separately. Now however, he adds, 
the proposition is proved generally. Yet Euclid says nothing to connect 
the two theories of proportion even when he comes in x. 5 to a proportion 
two terms of which are magnitudes and two are numbers (" Com mensurable 
magnitudes have to one another the ratio which a number has to a number"). 
The probable explanation of the phenomenon is that Euclid simply followed 
tradition and gave the two theones as he found them. This would square 
with the remark in Pappus (vii. p. 678) as to Euclid's fairness to others and 
his readiness to give them credit for their work. 



DEFINITIONS. 

1. A magnitude is a part of a magnitude, the less of 
the greater, when it measures the greater. 

2. The greater is a multiple of the less when it is 
measured by the less. 



114 ■ ' ' BOOK V ' [v. DKFF, 3—13 

3. A ratio is a sort of relation in respect of size between 

two magnitudes of the same kind. . , 

4. Magnitudes are said to have a ratio to one another 
which are capable, when multiplied, of exceeding one another. 

5. Magnitudes are said to be in the same ratio, the 
first to the second and the third to the fourth, when, if any 
equimultiples whatever be taken of the first and third, and 
any equimultiples whatever of the second and fourth, the 
former equimuhiples alike exceed, are alike equal to, or alike 
fall short of, the latter equimultiples respectively taken in 
corresponding order. 

6. Let magnitudes which have the same ratio be called 
proportional. 

7. When, of the equimultiples, the multiple of the first 
magnitude exceeds the multiple of the second, but the multiple 
of the third does not exceed the multiple of the fourth, then 
the first is said to have a greater ratio to the second than 
the third has to the fourth. 

8. A proportion in three terms is the least possible. 

9. When three magnitudes are proportional, the first is 
said to have to the third the duplicate ratio of that which 
it has to the second. 

10. When four magnitudes are < continuously > propor- 
tional, the first is said to have to the fourth the triplicate 
ratio of that which it has to the second, and so on con- 
tinually, whatever be the proportion. 

11. The term corresponding magnitudes is used of 
antecedents in relation to antecedents, and of consequents in 
relation to consequents. 

12. Alternate ratio means taking the antecedent in 
relation to the antecedent and the consequent in relation to 
the consequent. 

13. Inverse ratio means taking the consequent as 
antecedent in relation to the antecedent as consequent. 



V. DEFF.] DEFINITIONS 115 

14. Composition of a ratio means taking the ante- 
cedent together with the consequent as one in relation to 
the consequent by itself. 

15. Separation of a ratio means taking the excess 
by which the antecedent exceeds the consequent in relation 
to the consequent by itself. 

16. Conversion of a ratio means taking the ante- 
cedent in relation to the excess by which the antecedent 
exceeds the consequent. 

r< 17, A ratio ex aequali arises when, there being several 
magnitudes and another set equal to them in multitude which 
taken two and two are in the same proportion, as the first is 
to the last among the first magnitudes, so is the first to the 
last among the second magnitudes ; 

Or, in other words, it means taking the extreme terms 
by virtue of the. removal of the intermediate terms. 

18. A perturbed proportion arises when, there being 
three magnitudes and another set equal to them in multitude, 
as antecedent is to consequent among the first magnitudes, 
so is antecedent to consequent among the second magnitudes, 
while, as the consequent is to a third among the first 
magnitudes, so is a third to the antecedent among the second 
magnitudes. 

Definition i. 

The word/ar/ (fii'pot) is here used in the restricted sense of a submtiitipU 
or an aliquot part as distinct from the more general sense in which it is used 
in the Common Notion (;) which says that "the whole is greater than the 
part." It is used in th^ same restricted sense in vii. Def, 3, which is the same 
definition as this with "number" (opifl/iot) substituted for "magnitude." 
VII. Def. 4, keeping up the restriction, says that, when a number does not 
measure another (>umoer, it is farfs (in the plural), not b part of it. Thus, 
I, a, or 3, is a part of 6, bat 4 is not a pari of 6 but parts. The same 
distinction between the restricted and the more general sense of the word 
part appears in Aristotle, Mdaph. 1023 b is: "In one sense a part is 
that into which quantity (to irocrov) can anyhow be divided ; for that which is 
taken away from quantity, guA quatitity, is always called a 'part' of it, as 
e.g. two is said to be in a sense a part of three. But in another sense a 
'part' ill only what mtasura (ra Karo/wTpovtra) such quantities. Thus two 
b in one sense said to be a part of three, in the other not." 



.%>t6 BOOK V [v. DEFF. a, 3 

1. r - Definition 2. *< ►/r* 

noXXa)rX(i(rtov Si to ^«i£oy roC lAttrnivaft orov narajitTp^Tai vn> raC 
IXaTTOfot. 

Definition 3. 

AvytK ivrl Suo /iicyfPur ofiOytv^if ^ Kara m^Xifcon^ra vota <r^w($. 

The best explanation of the definitions of ratio miA proportion that I have 
seen is that of De Moigan, which will be found in the articles under those 
titles in the Penny Cyclopaedia, Vol xix. (1841) ; and in the following notes 
I shall draw largely from these articles. Very valuable also aie the notes on 
the definitions of Book v. given by Hanlcel (fragment on Euclid published as 
an appendix to his work Zur GeschiehU der Mathimaiik in AUtrthum und 
Mittdalier, 1874). 

There has been controversy as to what is the proper translation of the 
word in)Xucar)}s in the definition, irxitrit 'ara n-i^XtxcTTifTii has generally been 
translated " relation in respect of quantify." Upon this De Morgan remarks 
that it makes nonsense of the definition ; "for magnitude has hardly a 
different meaning from quantity, and a relation of magnitudes with respect to 
quantity may give a clear idea to those who want a word to convey a notion 
of architecture with respect to building or of battles with respect to fighting, 
and to no others." The true interpretation De Morgan, following Wallis and 
Gregory, takes to be guantuplidty, referring to the number of times one 
magnitude is contained in the other. For, he says, we cannot describe 
magnitude in language without quantuplicitative reference to other magni- 
tude; hence he supposes that the definition simply conveys the fact that the 
mode of expressing quantity in terms of quantity is entirely based upon the 
notion of quantuphcity or that relation of which we take cognizance when we 
find how many times one is contained in the other. While all the rest of 
De Morgan's observations on the definition are admirable, it seems to me 
that on ttiis question of the proper translation of infAtKo'Tijt he is in error. He 
supports his view tnainly by reference (i) to the definition of a compounded 
ratio usually given as the 5th definition of Book vi., which speaks of the 
TiiKiKonfrti of two ratios being multiplied together, and (t) to the comments 
of Eutocius and a scholiast on this definition. Eutocius says namely 
(Archimedes, ©d. Heiberg, iii, p. wo) that "the term njKuainp is evidently 
used of the number from which the given ratio is called, as (among others) 
Nicomachus says in his first book on music and Heion in his commentary 
on the Introduction to Arithmetic." But it now appears certain that this 
definition is an interpolation ; it is never used, it is not found in Campanus, 
and Peyrard's MS. only has it in the margin. At the same time it is clear 
that, if the definition is admitted at all, any commentator would be obliged to 
explain it in the way that Eutocius does, whether the explanation was consistent 
with the proper meaning of mfXutorijv or not. Hence we must look elsewhere 
for the meaning of m^XiKot and nrXtitttnTt. If we do this, I think we shall find 
no case in which the words have the sense attributed to them by De Morgan. 
The teal meaning of irufXi'itos is how great. It is so used by Aristotle, e.g. in 
Eth, Me. V. to, 1134 b 11, where he speaks of a man's child being as it were 
a part of him so long as he is of a certain age (lutt av ^ m^XtKcn'), Ag»in 
Nicomachus, to whom Eutocius appeals, himself (i, 2, 5, p. Si ed. Hoche) 
distinguishes ttjjXikov as referring to magnitude, while h-octo's refers to multitude. 
So does lamblichus in his commentary on Nicomachus (p. 8, 3 — 5) ; besides 
which lamblichus distinguishes irq^dKov as the subject of geometry, being am- 



V. DEF. 3] NOTE ON DEFINITION 3 117 

tinucus, and irocraf as the subject of arithmetic, being discrele, and speaks of a 
point being the origin of Tnjkinov as a unit is of iroo-oV, and so on. Similarly, 
Ptolemy (Syntaxis, ed, Heiberg, p. 31) speaks of the sise (injXijtdr)^) of the 
chords in a circle (jr<pi njt mjXntoTifros tw jv rifi kuhXj^ ti6nwv). Consequently 
I think we can only translate wii\tK6n)t in the definition as size. This 
corresponds to Hankel's translation of it as " GrOsse," though he uses this 
same word for a concrete " magnitude " as well ; size seems to me to give 
the proper distinction between injXiitontt and ii,iyt0o^, as size is the attribute, 
and a magnitude {in its ordinary mathematical sense) is the thing which 
possesses the attribute of siie. 

The view that " relation in respect of iize " is meant by the words in the 
text is also confirmed, I think, by a later remark of De Morgan himself, 
tiamely that a synonym for the word raiis may be found in the more in- 
telligible term relative magniiude. In fact axvm in the definition corresponds 
to relative and in/XotoTj)^ to magnitude. (By magnitude De Mo^an here 
means the attribute and not the thing possessing it.) 

Of the definition as a whole Simson and Hankel express the opinion that 
it is an interpolation. Hankel points to the fact that it is unnecessary and 
moreover so vague as to be of no practical use, while the very use of the 
expression na™ mjXtitorijTa seems to him suspicious, since the only other 
place in which the word wrjkiKoxTp occurs in Euclid is the 5th definition of 
Book VI., which is admittedly not genuine. Yet the definition of ratio appears 
in all the MSS., the only variation being that some add the words npm oAAifXo, 
"to one another," which are rejected by Heiberg as an interpolation of 
Theon ; and on the whole there seems to be no sufficient ground for regarding 
it as other than genuine. The true explanation of its presence would appear 
to be substantially that given by Barrow {Lectiones Cantabrig., London, 1684, 
Lect. Ill, of 1666), namely that Euclid bserted it for completeness' sake, mote 
for ornament than for use, intending to give the learner a general notion of 
ratio by means of a metaphysical, rather than a mathematical definition ; " for 
metaphysical it is and not, properly speaking, mathematical, since nothing 
depends on it or is deduced from it by mathematicians, nor, as I think, can 
anything be deduced." This is confirmed by the fact that there is no 
definition of Xo'yot in Book vii., and it could equally have been dispensed 
with here. Similarly De Morgan observes that Euclid never attempts this 
vague sort of definition except when, dealing with a well-known term of 
common life, he wishes to bring it into geometry with something like an 
expressed meaning which may aid the conception of the thing, though it does 
not furnish a perfect criterion. Thus we may compare the definition with 
that of a straight line, where Euclid merely calls the reader's attention to the 
well-known term tiStia ypa^/iij, tries how far he can present the conception 
which accompanies it in other words, and trusts for the correct use of the 
term to the axioms (or postulates) which the universal conception of a straight 
line makes self-evident. 

We have now to trace as clearly as passible the development of the 
conception of Xcfyoi, ratio, or relative magnitude. In its primitive sense 
Xff/os was only used of a ratio between com mensu rabies, i.e. a ratio which 
could be expressed, and the manner of expressing it is indicated in the 
proposition, Eucl. x. 5, which proves that commensurate magnitudes have to 
one another the ratio whieh a numl>er has to a number. That this was the 
primitive meaning of Aoyoi is proved by the use of the term uAoyoi for the 
mcom mensurable, which means irrational in the sense of not having a ratio 
to something taken as rational (^TTot). ^, , , . 



\t)m I ^^^ BOOK V •!''•' [v. DEF. 3 

- 1 Euclid himself shows us how we are to set about finding the ratio, or 
relative magiiitude, of two commensurable magnitudes. He gives, in x. 3, 
practically our ordinary method of finding the greatest common measure. 
If ^, ^ be two magnitudes of which B h the less, we cut off from A a part 
equal to B, from the remainder a part equal to B, and so on, until we leave a 
remainder less than B, say Ji,. We measure off ^, frcn S in the same way 
until a remainder X., is left which is less than fii- We repeat the process 
with ^1, B,, and so on, until we find a remainder which is contained in the 
preceding remainder a certain number of times exactly. If account is taken 
of the number of times each magnitude is contained (with something over, 
except at the last) in that upon which it is measured, we can calculate how 
many times the last remainder is contained in A and how many times the 
last remainder is contained in B ; and we can thus express the ratio of A to 
B as the ratio of one number to another. 

But it may happen that the two m^^ttudes have no common measure, 
i.e. are incommensurable, in which case the process described would never 
come to an end and the means of expression would fail ; the magnitudes 
would then Aave na ratio in the primitive sense. But the word Aoyos, ratio, 
acquires in Euclid, Book v., a wider sense covering the relative magnitude of 
incommensurabies as well as commensurables ; as stated in Euclid's 4th 
definition, "magnitudes are said to have a ratio to one another which can, 
when multiplied, exceed one another," and finite incommensurabies have this 
property as much as commetisurables. De Morgan explains the manner of 
transition from the narrower to the wider signification of ratio as follows, 
"Since the relative magnitude of two quantities is always shown by the 
quantuplicitative mode of expression, when that is possible, and since pro- 
portional quantities (pairs which have the same relative magnitude) are pairs 
which have the same mode {if possible) of expression by means of each other ; 
in all such cases sameness of relative magnitude leads to sameness of mode of 
expression ; or proportion is sameness of ratios (in the primitive sense). But 
sameness of relative magnitude may exist where quantuplicitative expression 
is impossible ; thus the diagonal of a larger square is the same compared with 
its side as the diagonal of a smaller square compared with its side. It is an 
easy transition to speak of sameness of ratio even in this case ; that is, to use 
the term ratio in the sense of relative magnitude, that word having originally 
only a reference to the mode of expressing relative magnitude, in cases which 
allow of a particular mode of expression. The word irraiional (SXsrpsi) does 
not make any corresponding change but continues to have its primitive 
meaning, namely, incapable of quantuplicitative expression." 

It remains to consider how we are to describe the relative magnitude of 
two incommensurabies of the same kind. That they have a definite relation 
is certain. Suppose, for precision, that S is the side of a square, D its 
diagonal ; then, if .S is given, any alteration in D or any error in D would 
make the figure cease to be a square. At the same time, a person altogether 
ignorant of the relative magnitude of D and 5 might say that drawing two 
straight lines of length .S so as to form a right angle and joining the ends by 
a straight line, the length of which would accordingly be D, does not help 
him to realise the relative magnitude, but that he would like to know how 
many diagonals make an exact number of sides. We should have to reply 
that no number of diagonals whatever makes an exact number of sides ; but 
that he may mtaition any fraction of the side, a hundredth, a thousandth or 
a millionth, and that we will then express the diagonal with an error not so 
great as that fraction. We then teU him that 1,000,000 diagonals exceed 



T. DW. j] NOTE ON DEFINITION 3 119 

1,414,113 sides but fall short of 1,414,214 sides; consequently the diagonal 
lies between t '41 41 13 and i '4 142 14 times the side, and these differ only by 
one-millionth of the side, so that the error in the diagonal is less still. To 
enable him to continue the firocess further, we show him how to perform the 
arithmetical operation of approximating to the value of J 2. This gives the 
means of carrying the approximation to any degree of accuracy that may be 
desired. In the power, then, of carrying approximations of this kind as far as 
we please lies that of expressing the ratio, so far as expression is possible, and 
of comparing the ratio with others as accurately as if expression had been 
possible^ 

Euclid was of course aware of this, as were probably others before him ; 
though the actual approximations to the values of ratios of incommensurabies 
of which we find record in the works of the great Greek geometers are very 
few. The history of such approximations up to Archimedes is, so far as 
material was available, sketched in 7%e Works of Archimides (pp. Ixxvti and 
following); and it is sufficient here to note the facts (i) that Plato, and,e*'en 
the PythagoreaiK, were familiar with J as an approximfvtion to .j^, {2) that 
the method of finding any number of successive approximations by the system 
of side- and iftajfo/taZ-numbere described by Theon of Smyrna was also 
Pythagorean (cf. the note above on Euclid, n. 9, 10), (3} that Archimedes, 
without a word of preliminary «tplanation, gives out that 

gives approximate values for the square roots of several large numbers, and 
proves that the ratio of the circumference of a circle to its diameter is less 
than 3t but greater than j-rii (4) '^^t the first approach to the rapidity with 
which the decimal system enables us to approximate to the value of surds 
was furnished by the method of sexagesimal fractions, which was almost as 
convenient to work with as the methoid of decimals, and which appears fully 
developed in Ptolemy's avyra^vi. A number consisting of a whole number 
and any fraction was under this system represented as so many units, so 
many of the fractions which we should denote by ^^, so many of those which 
we should write (jj)', (A)'> *"<1 ^° ""■ Theon of Alexandria shows us how 
to extract the square root of 4500 in this sexagesimal system, and, to show 
how effective it was, it is only necessary to mention that Ptolemy gives 

-—5 + ^j + ~ as an approximation to ^3, which approximation is equi«ilent 

to 17320509 in the ordinary decimal notation and is therefore correct to 
6 places. 

Between Def. 3 and Def, 4 two manuscripts and Campanus insert " Pro- 
portion is the sameness of ratios" (avoAoyta St jJ ruf AoyuiK rnvTonft), and even 
the best ms. has it in the margin. It would be altogether out of place, since 
it is not till Def, 5 that it is explained what sameness of ratios is. The words 
are an interpolation later than Theon (Heiberg, Vol. v, pp. xxxv, Ixxxix), 
and are no doubt taken from arithmetical works {cf Nicomachus and Theon 
of Smyrna). It is true that Aristotle says similarly, " Proportion is equality 
of ratios" (Eth. Nic. v. 6, 1131 a 31), and he appear to be quoting from 
the Pythagoreans ; but the reference is to numbers. 

Similarly two mss. (inferiorX insert after Def 7 "Proportion is the similarity 
{^^juartp) of ratios." Here too we have a mere interpolation. 



130 • BOOK V ' • [v. DEFF. 4,5 

Definition 4. 

\iymr ^f(v irpot SXXTjXa. furftdTj Kiytna, A St/rarai inAkiarXiuria^iiiMXi 

This definition supplements the last one. De Morgan says that it amounts 
to saying that the magnitudes are of the same species. But this can hardly 
be aU ; the definition seems rather to be meant, on the one hand, to exclude 
the relation of a finite magnitude to a magnitude of the same kind which is 
either infinitely great or infinitely small, and, even more, to emphasise the 
fact that the term ra/t'e, as defined in the preceding definition, and about to 
be used throughout the book, includes the relation between any two t'ncom- 
nunsurable as well as between any two commensurable finite m^nitudes of 
the same kind. Hence, while De Morgan seems to regard the extension of 
the meaning of ratio to include the relative magnitude of incommensurables 
as;, so to speak, taking place between Def. 3 and I>ef. Si the 4th definition 
appears to show that it is ratio in its extended sense that is being defined in 
Def. 3- . 

Definition S' 

TrfapTOf, otar to, toC TpajTOU KoX TptVov laaKK iroAAairAotrio tw toB Sniripoo 
xot rtrdpfTov utokk TroKXwrXairitiiv Kau ottoiovovv TroXXairAao'UKr/io^ iKartpcv 

In my translation of this definition I have compromised between an 
attempted literal translation and the more expanded version of Simson. 71ie 
difiiculty in the way of an exactly literal translation is due to the fact that the 
words (KaS* ijToiovoIi' jroXAaTrXao-iarr/iov) signifying that the equimultiples in 
eiuh ease are any equimultiples wAa/evcr occur only once in the Greek, though 
they apply ieiA to Ta....'uTdKK wokkan-kajria in the nominative and Tuv...Uri*is 
u-oXAmrAno-i'ui' in the genitive. I have preferred "alike " to " simultaneously" 
as a translation of a/ia because " simultaneously " might suggest that time was 
of the essence of the matter, whereas what is meant is that any particular 
comparison made between the equimultiples must be made between (At same 
equimultiples of the two pairs respectively, not that they need to be compared 
at the same time, 

Aristotle has an allusion to a definition of " the same ratio " in Tcfiia 
VIII. 3, 158 b 29 ; " In mathematics too some things appear to be not easy to 
prove {ytid^fa6ai) for want of a definition, e.g. that the parallel to the side 
which cuts a plane [a parallelogram] divides the straight hne [the other side] 
and the area similarly. But, when the definition is expressed, the said property 
is immediately manifest ; for the areas and the straight lines Aave the same 
di'Tai'tn'p«ri9, and this is the definition of 'the same ratio.'" Upon this 
passage Alexander says similarly, " This is the definition of proportionals 
which the ancients used : magnitudes are proportional to one another which 
have (or show) ihe same mSv^tupttrvst and Aristotle has called the latter 
ivravalpKrit." Heiberg (Mathematisehes zu Aristofeles, p. 2 a) thinks that 
Aristotle is alluding to the fact that the proposition referred to could not be 
rigorously proved so long as the Pythagorean definition applicable to com- 
mensurable magnitudes only was adhered to, and is (quoting the definition 
belonging to the complete theory of Eudoxus ; whence, m view of the positive 
statement of Aristotle that the definition quoted is the definition of "the same 
ratio," it would appear that the Euclidean definition (which Heiberg describes 
as a careful and exact paraphrase of d-vTovalpftrn) is Euclid's own. I do not 



V. DEF. s] NOTES ON DEFINITIONS 4, 5 t»l 

feel able to subscribe to this view, which seems to me to involve very grave 
difiScuUies. The Euclidean definition is regularly appealed to in Book v. as 
the criterion of magnitudes being in proportion, and the use of it would appear 
to constitute the whole essence of the new general theory of proportion; if then 
this theory is due to Eudoxus, it seems impossible to believe that the definition 
was not also due to him. Certainly the definition given by Aristotle would 
be no substitute for it; dvfiv4>aipft7K and dnavoiptaK are words almost a:! 
vague and " metaphysical " (as Barrow would say) as the words used to define 
raifa, and it is difficult to see how any mathematical facts could be deduced 
from such a definition. Consider for a moment the etymology of the words. 
w^oifKo-ts or dva^ttFii means " removal," " taking away "or " destruction " of 
a thing; and the prefix om indicates that the "taking away" from one 
magnitude answers to, corresponds with, alternates with, the " taking away " 
from the other. So fai» therefore as the etymology goes, the word seems 
rather to suggest the " taking away " of corresponding fractions, and therefore 
to suit the old imperfect theory of proportion rather than the new one. Thus 
Waitz {ad lac.) paraphrases the definition as meaning that " as many parts as 
are taken from one magnitude, so many are at the same time taken from the 
other as well," A possible explanation would seem to be that, though 
Eudoxus had formulated the new definition, the old one was still current in 
the text-books of Aristotle's time, and was taken by him as being a good 
enough illustration of what he wished to bring out in the pas.sage of the 
Ibpia referred to. 

From the revival of learning in Europe onwards the Euclidean definition 
of proportion was the subject of much criticism. Campanus had failed to 
understand it, had in fact misinterpreted it altogether, and he may have 
misled others such as Ramus {1515 — 72), always a violently hostile critic of 
Euclid. Among the objectors to it was no less a person than Galileo. For 
particulars of the controversies on the subject down to Thomas Simpson 
\Elem. of Geometry, Lond. i8oo) the reader is referred to the Excursus at the 
end of the second volume of Camerer's Euclid (1825). For us it is interesting 
to note that the unsoundness of the usual criticisms of the definition was 
never better exposed than by Barrow. Some of the objections, he pointed out 
{tttt. Cantabr. vn.ofi665),areduetom isconception onthepartoftheir authors 
as to the nature of a definition. Thus Euclid is required by these objectors 
(e.g. Tacquet) to do the impossible and to show that what is predicated in the 
definition is true of the thing defined, as if any one should be required to 
show that the name "circle" was applicable to those figures alone which 
have their radii all equal ! As we are entitled to assign to such figures and 
such figures only the name of "circle," so Euclid is entitled {"quamvis non 
temere nee imprudenter at certii de causis iustis illis et idoneis") to describe 
a certain property which four magnitudes may have, and to call magnitudes 
possessing that property magnitudes "in the same ratio." Others had argued 
from the occurrence of the other definition of proportion in vii. Def. so that 
Euclid was dissatisfied with the present one ; Barrow pointed out that, on the 
contrary, it was the fact that vu. Def. 3o was not adequate to cover the case 
of incommensurables which made Euclid adopt the present definition here. 
Lastly, he maintains, gainst those who descant on the "obscurity" of v. 
Def. 5, that the supposed obscurity is due, partly no doubt to the inherent 
difficulty of the subject of incommensurables, but also to faulty translators, 
and most of all to lack of effort in the learner to grasp thoroughly the meaning 
of words which, in themselves, are as clearly expressed as they could be. 

To come now to the merits of the case, the best defence and explanation 



tss 



.1' 



BOOK V 



tA\-tV 



[v. DEF. S 



of the definition that I have seen is that given by De Morgan, He first 
translates it, observes that it applies equally to commensurable or incom- 
mensurable quantities because no attempt is made to measure one by an 
aliquot part of another, and then proceeds thus, 

"The two questions which must be asked, and satisfactorily answered, 
previously to its [the definition's] reception, are as follows : 

1. What right had Euclid, or any one else, to expect that the preceding 
most prolix and unwieldy statement should be received by the beginner as 
the definition of a relation the perception of which is one of the most common 
acts of his mind, since it is performed on every occasion where similarity or 
dissimilarity of figure is looked for or presents itself F 

2. If the preceding question should be clearly answered, how can the 
definition of proportion ever be used ; or how is it possible to compare every 
one of the infinite number of multiples of ji with every one of the multiples 
of^? 

To the first question we reply that not only is the test proposed by 
Euclid tolerably simple, when more closely examined, but that it is, or might 
be made to appear, an easy and natural consequence of those (iandamental 
perceptions with which it may at first seem difficult to compare it." 

To elucidate this De Morgan gives the following illustration. 

Suppose there is a straight colonnade composed of equidistant columns 
{which we will understand to mean the vertical lines forming the axes of the 
columns respectively), the first of which is at a distance from a bounding wall 
equal to the distance between consecutive columns. In front of the colonnade 
let there be a straight row of equidistant railings (regarded as meaning their 
axes), the first being at a distance from the bounding wall equal to the 
distance between consecutive railings. Let the columns be numbered from 
the wall, and also the railings. We suppose of course that the column distance 
(say, C) and the railing distance (say, Ji) are different and that they may bear 
to each other any ratio, commensurable or incommensurable ; i.e, that there 
need not go any exact number of railings to any exact number of columns. 



I \t a 4 B 6 T 8 



fl 10 tl 12 la 14 tfl la 17 la 



If the construction be supposed carried on to any extent, a spectator can, 
by mere inspection, and without measurement, compare C with Ji to any 
degree of accuracy. For example, since the loth railing falls between the 4th 
and 5th columns, 10^ is greater than 4C and less than $C, and therefore Jl 
lies between -yVhs of C and yjjths of C. To get a more accurate notion, the 
ten-thousandth railing may be talcen ; suppose it falls between the 4674th and 
4675th columns. Therefore io,ooo.ff lies between 46 74 C and 4675 C, or ^ hes 
between tVuVTy ^'^^ rVtyVs ^^ ^' There is no limit to the degree of accuracy 
thus obtainable ; and the ratio of ^ to C is determined when the order of 
distribution of the railings among the columns is assigned arf infinitum ; or, in 
other words, when the position of any giver railing can be found, as to the 
numbers of the columns between which it lies. Any alteration, however 
small, in the place of the first railing must at last affect the order of 
distribution. Suppose e.g. that the first railing is moved from the wall by one 
part in a thousand of the distance between the columns ; then the second 
railing is pushed forward by x7n(irC, the third by nnnr^i and so on, so that 



V. DEF, 5] NOTE ON DEFINITION 5 i«3 

the railings after the thousandth are pushed forward by more than C; i.e. the 
order with respect to the columns is disarranged. 

Now let it be proposed to make a model of the preceding construction in 
which c shall be the column distance and r the railing distance. It needs no 
definition of proportion, nor anything more than the conception which we 
have of that term prior to definition {and with which we must show the agree- 
ment of any definition that we may adopt), to assure us that C must be to J 
in the same prof>ortion as ^ to r if the model be truly formed. Nor is it 
drawing too largely on that conception of proportion to assert that the 
distribution of the railings among the columns in the model must be every- 
where the same as in the original ; for example, that the model would be out 
ef proportion if its 37th railing fell between the i8th and 19th columns, while 
the 37th rathng of the original fell between the 17th and iSth columns. Thus 
the dependence of EucHd's definition upon common notions is settled; for the 
obvious relation between the construction and its model which has just been 
described contains the collection of conditions, the fulfilment of which, 
according to Euclid, constitutes proportion. According to Euclid, whenever 
mC exceeds, equals, or falls short of nR, then tttc must exceed, equal, or fall 
short of nr; and, by the most obvious property of the constructions, according 
as the wth column comes after, opposite to, or before the nth railing in the 
original, the ffith column must come after, opposite to, or before the ffth 
railing in the correct model. 

Thus the test proposed by Euclid is necessary. It is also sufficient. For 
admitting that, to a given original with a given column-distance in the model, 
there is one correct model railing distance (which must therefore be that 
which distributes the railings among the columns as in the original), we have 
seen that any other railing distance, however slightly different, would at last 
give a difTerent distribution ; that is, the correct distance, and the correct 
distance only, satisfies all the conditions required by Euclid's definition. 

The use of the word diitribtition having been well learnt, says De Morgan, 
the following way of stating the definition will be found easier than that of 
Euclid. " Four magnitudes, A and B of one kind, and C and D of the same 
or another kind, are proportional when all the multiples of A can be 
distributed among the multiples of Bm the same intervals as the correspond- 
ing multiples of C among those of D." Or, whatever numbers m, n may be, 
if mA lies between ttB and {n + i)B, mC lies between nD and (« + i)£>. 

It is important to note that, if the test be always satisfied from and after 
any given multiples of A and C, it must be satisfied before those multiples. For 
instance, let the test be always satisfied from and after \oaA and looC; and 
let f)A and 5C be instances for examination. Take any multiple of 5 which 
will exceed 100, say 50 times five ; and let it be found on examination that 
250^4 lies between 678^ and 67g.fi ; then 150 c lies between 678ZJ and 
f>1^D. Divide by 50, and it follows that ^A lies between \%\%B and tzWB, 
and ajortiori between i^B and \^B. Similarly, 5 dies between isl^^Z? and 
13JJ/), and therefore between \%D and 14Z?. Or ^A lies in the same 
interval among the multiples of B in which 5 C lies among the multiples of D, 
And so for any multiple of A, C less than 100^, looC. 

There remains the second question relating to the infinite character of the 
definition ; four magnitudes A, B, C, D are not to be called proportional 
until it \b shown that every multiple of A falls in the same intervals among 
the multiples of 5 in which the same multiple of C is found among the 
multiples of D. Suppose that the distribution of the raihngs among the 



'^''' BOOK V 



[v. DEF, S 



columns should be found to agree in the model and the original as far as 
the millionth railing. This proves only that the railing distance of the model 
does not err by the millionth part of the corresponding column distance. We 
can thus fix limits to the disproportion, if any, and we may make those limits 
as small as we please, by carrying on the method of observation; but we 
cannot obscrue an infinite number of cases and so enable ourselves to affirm 
proportion absolutely. Mathematical methods however enable us to avoid 
the difficulty. We can take any multipks whatever and work with them as if 
they were particular multipJes. De Morgan gives, as an instance to show that 
the definition of proportion can in practice be used, notwithstanding its 
infinite character, tiie following proof of a proposition to the same effect as 
EucL VI. 3. 




o. A, oj At 



"Let OAB be a triangle to one side AB of which ab is drawn parallel, and 
on OA produced set off A At, A^Af etc. equal to OA, and aa^ a,£i, etc. equal 
to Oii. 

Through every one of the points so obtained draw parallels to AB, 
meeting OB produced in b^, B, etc. 

Then it is easily proved that W„ bj^, etc. are severally equal to Ob, and 
BB^, B^Bi etc. to OB. 

Consequently a distribution of the multiples of OA among the multiples 
of Oa is made on one line, and of OB among those of Ob on the other. 

The examination of this distribution in all its extent (which is impossible, 
and hence the apparent difficulty of using the definition) is rendered 
unnecessary by the known property of parallel lines. For, since At lies 
between a, and a„ B^ must he between b^ and ^,j for, if not, the line A^B^ 
would cut either a^, or a^^. 

Hence, without inquiring where A,^ doei fall, we know that, if it fall 
between a, and a,^„ B,^ must fall between b„ and ^,+1 ; or, if m . OA fall in 
magnitude between n.Oa and (n + i)(?a, then m.OB must fall between 
n.Ob and («+i)Oi." 

Max Simon remarks {Euclid und die seeks planimeirischen Buchtr, p. no), 
after Zeuthen, that Euclid's definition of equal ratios is word for word the 
same as Weierstrass' definition of equal numbers. So far from agreeing in 
the usual view that the Greeks saw in the irrational no number, Simon thinks 
it is clear from Eucl. v. that they possessed a notion of number in all its 
generality as clearly defined as, nay almost identical with, Weierstrass' con- 
ception of it 

Certain it is that there is an exact correspondence, almost coincidence, 
between Euclid's definition of equal ratios and the modern theory of irrationals 
due to Dedekind. Premising the ordinal arrangement of natural numbers in 
ascending order, then enlarging the sphere of numbers by including 
(i) negative numbers as well as positive, (2) fractions, as ajb, where a, b may 



V. DEF. i] NOTE ON DEFINITION s itj 

be any natural numbers, provided that i is not zero, and arranging the 
fractions ordinally among the other numbers according to the definition : 

let 1 be < = > J according as a;/ is < = > ii:, 
e a 

Dedekind arrives at the following definition of an irrational number. 

An inatianal number a is defined whenever a law is stated which will 
assign every given rational number to one and only one of two classes A and 
B such that (i) every number in A precedes every number in j5, and (2) there 
is no last number in A and no first number in B ; the definition of a being 
that it is the one number which lies between all numbers in A and all 
numbers in B. 

Now let xly and ar'/y be equal ratios in Euclid's sense. 

Then ~ will divide all rational numbers into two groups A and B ; 
—, „ „ „ A' and B". 



Let -; be any rational number in A, so that 



tax ,. 

This means that ay <bx. 

But Euclid's definition asserts that in that case af-cbs! also. 

Hence also 7 < -1 ; 

b y 

therefore every member of group A is also a member of group A'. 
Similarly every member of group ^ is a member of group B". 

For, if T belong to £, 

ax 

which means that ay > bx. 

But in that case, by Euclid's definition, «y > bx' ; 

therefore also i> -j- 

y 

Thus, in other words, A and B are coextensive with A' and S 
respectively ; 

therefore - = — , according to Dedekind, as well as according to Euclid. 

If x(y, :^iy happen to be rational, 
then one of the groups, say A, includes xjy, 
and one of the groups, say A', includes x'jy'. ' • 

d . X •■!.,-"' 

In this case r might mncide with - ; ,..-., 

X 

that is r = - 1 

b y' 

which means that ay^bx. 



i«6 ' '^^ BOOK V •' ' [v.DW. S 

Therefore, by Euclid's definition, ay = &e' ; -i-'wi-'* timu .,i.. •-■ 

SO that T^-5. 

y 

Thus the groups are again coextensive. 

In a woid, Euclid's definition divides all rational numbers into two 
coextensive classes, and therefore defines equal ratios in a manner exactly 
corresponding to Dedekind's theory. 

Alternativea for Eucl. V. Dcf. 5. 

Saccheri records in his Evclides ob omni noivo vindicatus that a distinguished 
geometer of his acquaintance proposed to substitute for Euclid's the following 
definition : 

"A first magnitude has to a second the same ratio that a third has to a 
fourth when the first contains the aliquot parts of the second, auording to any 
number [i.e. with any denominator] whatever, the same number of times as 
the number of times the third contains the same aliquot parts of the fourth " ; 
on which Saccheri remarks that he sees no advantage in this definition, which 
presupposes the notion of division, over that of Euclid which uses multiplication 
and the notions o{ greater, equal, and less. 

This definition was, however, practically adopted by Faifofer [Elementi it 
geometria, 3 ed., iSSsi) in the following form 1 

" Four infinitudes taken in a certain order form a proportion when, by 
measuring the first and the third respectively by any equi-submultiples 
whatever of the second and of the fourth, equal quotients are obtained," 

Ingrami {Elementi di geometria, 1904) takes multiples of the first and third 
instead of submultiples of the second and fourth : 

" Given four magnitudes in predetermined order, the first two homogeneous 
with one another, and likewise also the last two, the magnitudes are said to 
form a proportion (or to be in proportion) when any multiple of the first 
contains the second the same number of times that the equimultiple of the 
third contains the fourth." 

Veronese's definition {Elementi di geometria, PL 11., 1905) is like that of 
Faifofer; Enriques and Amaldi {Elemtnti di geom^ria, 1905) adhere to 
Euclid's: 

Proportionals of VII. Def. ao a particular case. 

It has already been observed that Euclid has nowhere proved (though the 
fact cannot have escaped him) that the proportion of numbers is included in 
the proportion of magnitudes as a special case. This is proved by Sim son as 
being necessary to the 5th and 6th propositions of Book x. Simson's proof is 
contained in his propositions C and D inserted in the text of Book v, and in 
the notes thereon. Proposition C and the note on it prove that, if four 
magnitudes are proportionals according to vii. Def. 20, they are also proportionals 
according to v. Def. 5. Prop, D and the note prove the partial converse, 
namely that, if four magnitudes are proportionals according to the Sth definition 
of Book v., and if the first be any multiple, or any part, or parts, of the second, 
the third is the same multiple, part, or parts, of the fourth. The proofs use 
certain results obtained in Book V. 

Prop. C is as follows ; 

If the first be the same multiple of the second, or the same part of it, that tht 
third is qfthe fourth, the first is to the second as the third to the fourth. 



V. DEF. s] NOTE ON DEFINITION s nf 

Let the (list A be the same multiple of B the second that C the third is of 
the fourth D \ 

^ is to ^ as C is to 27. 



A e 

B O 

C F 

D H ■ ' 

Take of A^ C any equimultiples whatever E, F\ and of B, D any 
equimultiple whatever G, H. 

Then, because -4 is the same multiple of B that C is of 27, 'i- " ; '" 
and E is the same multiple o{ A that F\% of C, 

E is the same multiple of B that F\% of D. [v, 3] 

Therefore E, E&k the same multiples of B, D. 

But G, H d,K equimultiples of ^, D; 

therefore, if £ be a greater multiple of B than G is, F'\% a greater multiple of 
JD than ^ is of i? ; 

that is, if £ be greater than G, Fis greater than If. 
In like manner, 

if E be equal to G, or less, ^is equal to J/, or less than it. 

But E, Fate equimultiples, any whatever, of ^, C; 
and G, H any equimultiples whatever of B, D. 

Therefore ^ is to ^ as C is to Z>. [v, Def. 5] 

Next, let the first A be the same pari of the second B that the third C is 
of the fourth D ; 

j4 is to i? as C is to ZJ. A 

For B is the same multiple of A that 2> is of C; B 

wherefore, by the preceding case, q 

^ is to j4 as /J is to C; O 

and, imxrsefy, A is to B as C is to D. 

[For this last inference Sirason refers to his Proposition B. That 
proposition is very simply proved by taking any equimultiples E, F of B, Z) 
and any equimultiples G, Hoi A, C and then arguing as follows : 

Since A\%to B && C is to Z>, 

G, If are simuUamously greater than, equal to, or less than E, F 
respectively ; so that 

E, F are HmultantauUy less than, equal to, or greater than G, H 
respectively, 

and therefore [Def. 5] ^ is to ^4 as i? is to C] 

We have now only to add to Prop. C the case where AB contains the 
same parts of CD that EFAaes of GH: 

in this case likewise AB is to CD as EFto GIf. 

Let CJC be a part of CD, and GL the same part of GIf; let AB be the 
same multiple of CAT that EFis of GL. 



iaS ' ^'- BOOK V [v. T>KF. 5 

Therefore, by Prop. C, '■"■ ■ '"• — - ■ 

A£ is to C^as £Fu> GL, 



B E- 

G- 



c— R ^ 

And CD, GH ire equimultiples of CK, GL, the second and fourth. 

Therefore AB is to CD as EF to G^ [Simson's Cor, to v. 4, which 
however is the particular case of V. 4 in which the " equimultiples " of one 
pair are the pair itself, i.e. the pair multiplied by unity]. 

To prove the partial converse we begin with Prop. D. 

If the first be to the second as the third to the fourth, and if the first be a 
multiple or part of the second, the third is the same mulliple or the same part of 
the fourth. 

Let v4 be to ^ as C is to i> ; 

and, first, let /i be a multiple of B ; 

C is the same multiple of S. 

Take E equal to A, and whatever multiple A 01 E \s of B, malce F the 
same multiple of Z>. 

Then, because A is Xo B a& C\s 10 D, <■' 

and of B the second and D the fourth equimultiples have been taken E 
and F, 

/i is to £ as C is to ^ [v. 4, Cor.] 

But A is equal to E ; 

therefore C is equd to F. 

[In support of this inference Simson cites his Prop. A, which however we 
can directly deduce from v. Def. 5 by taking any, but the same, equimultiples 
of all four magnitudes.] 



A C- 

B D- 

e— F- 



Now ,^is the same multiple of Z> that Aisol B; 

therefore C is the same multiple of D that A is of B. 
Next, let the first /i be a part of the second B ; 

C the third is the same part of the fourth D, 
Because ^i is to ,5 as C is to D, 

inversely, J is to -4 as Z) is to C. [Prop. B] 

But A'y&a. part of J; therefore ,5 is a multiple of >€; 

and, by the preceding case, D is the same multiple of C, 

that is, C is the same part of D that A is of B. 
We have, again, only to add to Prop D the case where AB contains any 
parts of CD, and AB is to CD as EFio GH; 

then shall EF zontain the same parts of GB that AB does of CD. 



?■ DWr. 5—7] NOTES ON DEFINITIONS g— 7 199 

For let CIC be a part of CJ), and G'Z the same part of Gff; and let j4£ 

be a multiple of CX. 

jff^ shall be the same multiple of GL. 

Take M the same multiple of GL that AB h of CA"; 
therefore ji£ is to CAT as ^ is to GL. [P"*p. C] 



A 


B 


E 




— F 


C tr- 





G- L 


H 


M 



And CI>, G/f&te equimultiples of CA", GL; 
therefore /4B is to CD as j}/ is to GIL 

But, by hypothesis, A£ is to CZJ as £^is to GB"; 

therefore M is equal to £J\ [v. 9] 

and consequently £^i3 the same multiple of GL that A£ is of Cff. 

Definition 6. ' 

t ■ 

Tb Si TOV airoy Ij^dktu \6yov /itye&Tj avaXjoyov KoXturBa, x > 

'A,vdXoryoy, though usually written in one word, is equivalent to aVi Xoyot', /» 
proportion. It comes however in Greek mathematics to be used practically as 
an indeclinable adjective, as here ; cf. oi Tto-o-opt? euflttdi avaktyfov laovtiu., 
" the four straight lines will be proportional," tflyw/a. rat irXm/Mi aVoAoyov 
Ix"**^ "triangles having their sides proportional." Sometimes it is used 
adverbially : akoXoyov o^ k^iv ci>c 17 BA irpo? r^v AF, outq^^ ?^ HA irpo^ T^f AZ, 
"proportionally therefore, as BA is to AC, so is CZJ to £>F"\ so too, ap- 
parently, in the expression ij /lAnj aniXoyoi/ (*ufl<ut), " the mean proportional." 
I do not follow the objection of Max Simon (Euclid, p. no) to "proportional" 
as a translation of oniAoyoi'. "We ask," he says, "in vain, what is proportional 
to what? We say e.g. that weight is proportional to price because double, treble 
etc. weight corresponds to double, treble etc. price. But here the meaning must 
be 'standing in a relation of proportion.'" Yet he admits that the Latin word 
preportionalis is an adequate expression. He transl.ttes by "in proportion" 
in the text of this definition. But I do not see that "in proportion " is better 
than "proportional." The fact is that both expressions are elliptical when 
used of four magnitudes " in proportion " ; but there is surely no harm in 
using either when the meaning is so well understood. 

The use of the word naXtiV^, " let magnitudes having the same ratio be 
called proportional," seems to indicate that this definition is Euclid's own. 



Definition 7. 

TOv Tov Btvripov iroAXa^rXaatou, to Sc to€ rpirov TroWaTrKa^ior ^ij {ijrtpi)fff rov 
ro3 rtropTou troXXavXairltni, rott to rpajrov irpoi to Stvrtpor fWifuwi Xoyoi" •X'"' 
Xfycrat, ^tp ra rpiVcn^ vpo^ to TtToprov. 

As De Moi^an observes, the practical test of disproportion is simpler than 
that of proportion. For, whereas no examination of individual cases, however 



130 ■ BOOK V ' [v. DBF. 7 

extensive, will enable an observer of the construction and its model (the 
illustration by means of columns and railings described above) to affinn 
proportion or deny disproportion, and all it enables us to do is to fix limits 
(as small as we please) to the disproportion (if any), a single instance may 
enable us to deny proportion or affirm disproportion, and also to slate which 
way the disproportion lies. Let the igth railing in the original fall beyond 
the nth column, while the 15th railing of the (so-called) model does not 
come up to the nth column. It follows from this one instance that the 
railing distance of the model is too small relatively to the column distance, or 
that the column distance is too great relatively to the railing distance. That 
is, the ratio of /• to r is less than that of /? to C, or the ratio of f to r is greater 
than that of C to jR. 

Saccheri (<7/. at,) remarks (as Commandinus had done) that the ratio of 
the first magnitude to the second will also be greater than that of the third to 
the fourth if, while the multiple of the first is efuai to the multiple of the 
second, the multiple of the third is Uss than that of the fourth : a case not 
mentioned in Euclid's definition. Saccheri speaks of this case being included 
in Clavius' interpretation of the definition. 1 have, however, failed to find a 
reference to the case in Clavius, though he adds, as a sort of corollary, in his 
note on the definition, that if, on the other hand, the multiple of the first is 
iess than the multiple of the second, while the multiple of the third is nt>i las 
than that of the fourth, the ratio of the first to the second is kss than that of 
the third to the fourth. 

Euclid presumably left out the second possible criterion for a greater ratio, 
and the definition of a less ratio, because he was anxious to reduce the 
definitions to the minimum necessary for his purpose, and to leave the rest to 
be inferred as soon as the development of the propositions of Book v. enabled 
this to be done without difficulty. 

Saccheri tried to reduce the second possible criterion for a greater ratio to 
that ^ven by Euclid in his definition without recourse to anything coming 
later in the Book, but, in order to do this, he has to use "multiples" produced 
by multipliers which are not integral numbers, but integral numbers //uJ proper 
fractions, so that Euclid's Def. 7 becomes inapplicable. 

De Morgan notes that " proof should be given that the same pair of 
magnitudes can never offer both tests [i.e. the test in the definition for a 
greater ratio and the corresponding test for a less ratio, with "less" substituted 
for "greater" in the definition] to another pair; that is, the test of greater 
ratio from one set of multiples, and that of less ratio from another." In other 
words, if m, n, p, q are integers and A, B, C, D four magnitudes, none of the 
pairs of equations 

(i) mA->HB, mC=ai <nD, 
(a) mA = nB, mC < nD 
can be satisfied simultaneously with any one of the pairs of equations 

(3) pA^qB, pC>qD, 

(4) pA < qB, y*C > or = qD. 

There is no difficulty in proving this with the help of two simple 
assumptions which are indeed obvious. 

We need only take in illustration one of the numerous cases. Suppose, if 
possible, that the following pairs of equations are simultaneously true : 

(l) viA>nB, mC<nD 
and (2) pA <qB, pC>qD. 



V. DEFF. r, 8] NOTES ON DEFINITIONS ;, 8 iji 

Multiply (i) by q and (z) by n. 

(We need here to assume that, whet« rX, rK are any equimultiples of any 
magnitudes X^ ¥, 

according as X>- = < Y, rX> = <rY. 

This is contained in Simson's Axioms i and 3.) 

We have then the pairs of equations 

my A > ngB, mqC < n^D, 

npA<nqB, npC>nqD, 

From the second equations in each pair it follows that 

mqC < npC. 

(We now need to assume that, if rX, sX are any multiples of X, and 
rY, sY the same multiples of Y, then, 

according as rX >-< sX, rY-> = < sY. 

Simson uses this same assumption in hb proof of v. i3.) 

Therefore mqA <npA, ■ , 

But it follows from the first equations in each pair that 

mqA > npA : 
which is impossible. 

Nor can Euclid's criterion for a greater ratio coexist with that for equal 

ratios. 



Definition 8. 

^ kvoXvfia, Si iv fptaXv opoit ISaxurryi hrriv. 

This is the reading of Heiberg and Camerer (who follow Peyrard's Ms,) 
and is that translated above. The other reading has Aaxiorcni, which can 
only be translated "consists in three terms a/ hast." Hankel regards the defi- 
nition as a later interpolation, because it is superfluous, and because the word 
ojxK for a term in a proportion is nowhere else used by Euclid, though it is 
common in later writers such as Nicomachus and Theon of Smyrna. The 
genuineness of the definition is however supported by the fact that Aristotle 
not only uses Spot in this sense {Eth. Nic. v. 6, 7, 1131 b 5, 9), but has a similar 
remark {ibid. 1131 a 31) that a "proportion is in fQiir terms at least" The 
difference from Euclid is only formal ; for Aristotle proceeds : " The diicrett 
(Si^p^fiini) (proportion) is clearly in four (terms), but so also is the continuous 
(awtxv'). For it uses one as two and mentions it twice, e.g. (in stating) that, 
as a is to j3, so also is j3 to y ; thus j3 is mentioned twice, so that, if /9 be twice 
put down, the proportionals are four." The disrinction between discrete and 
C07ttinuous seems to have been Pythagorean (cf. Nicomachus, 11. 11, 5; 23, 
a, 3; where however o-un/^fioTj is used instead of Tvvtxq^); Euclid does not 
use the words Stjpit|/t(n; and awtx^i in this connexion. 

So far as they go, the first words of the next definition (g), "When three 
magnitudes are proportionals," which seemingly refer to Def. 8, also support 
the view that the latter is, at least in substance, genuine. 



ij» *' BOOK V -'•:■' [v. DKFF. 9,10 



<■' 



•••••- Definitions 9, la t 

9. 'Oral' Si Tpi'a fityi^rj ivaXayov % to irpwroi' jrpos ri TpiTOf fcirXnirtova 
Xoyov <;(<£i^ Xcycrai tttc/i irput to BtvTtpov. 

10. 'Orair Sc THTO'a^ML fttyi&Tq aVaXoyoi^ ^^ to vfitarav irpm To rireifrTOV 
Tpurkatrlova Awyoy fx** '^ty^riM ^irep irpos li MvTtpoy, nai dtt if^t opHuii, wt 

Here, and fn connexion with the definitions of duplicate, triplicate, etc. 
ratios, would be the place to expecta definition of "compaunii ratio." None 
such is however forthcoming, and the only "definition" of it that we find is 
that forming vi. Def. Si which is an interpolation made, perhaps, even before 
Theon's time. According to the interpolated definition, " A ratio is said to 
be compounded of ratios when the sizes (jnfAtKo'nTret} of the ratios multiplied 
together make some (? ratio)." But the multiplication of the li^s (or 
magnitudes) of two ratios of incommensurable, and even of commensurable, 
magnitudes is an operation unknown to the classical Greek geometers. 
Eutocius (Archimedes, ed. Heiberg, iii. p. lao) is driven to explain the 
definition by making irijAwonj^ mean the number from which the given ratio 
is called, or, in other words, the number which multiplied into the consequent 
of the ratio gives the antecedent. But he is only able to work out his idea with 
reference to ratios between numbers, or between commensurable magnitudes ; 
and indeed the definition is quite out of place in Euclid's theory of 
proportion. 

There is then only one statement in Euclid's text as we have it indicating 
what is meant by compound ratio ; this is in vi: 23, where he says abruptly 
"But the ratio of KXa M is compounded of the ratio of JT to Z and that of 
L to M." Simson accordingly gives a defitiition (A of Book v.) of compound 
ratio directly suggested by the statement in vi. 23 just quoted. 

" When there are any number of magnitudes of the same kind, the first 
is said to have to the last of them the ratio compounded of the ratio which 
the first has to the second, and of the ratio which the second has to the third, 
and of the ratio which the third has to the fourth, and so on unto the last 
magnitude. 

For example, if A^ B, C, D b& four ma^itudes of the same kind, the 
first A is said to have to the last D the ratio compounded of the ratio of 
A to B, and of the ratio of B to C, and of the ratio of C to Z) ; or the ratio 
o^ A 10 Dm said to be compounded of the ratios of A ta S, B to C, and 
C to D. 

And if .(4 has to .ff the same ratio which E has to F; and .ff to C the 
same ratio that G has to /f ; and C to Z> the same that A" has to L; then, 
by this definition, A is said to have to D the ratio compounded of ratios 
which are the same with the ratios of E io F, G to H, and A" to Z : and the 
same thing is to be understood when it is more briefly expressed, by saying, 
A has to D the ratio compounded of the ratios of £■ to ^ G to If, and 
JT to Z, 

In like manner, the same things being supposed, if M has to N the 
same ratio which A has to D ; then, for shortness' sake, M is said to have to 
A^the ratio compounded of the ratios of -£■ to ^ G to H, and A" to L." 

De Morgan has some admirable remarks on compound ratio, which 
uot only give a very clear view of what is meant by it but at the same time 



V. vwr. 9, lo] NOTES ON DEFINITIONS 9, 10 133 

supply a. plausible explanation of the origin of the term. "Treat ratio," says 
De Morgan, ''as an engine of operation. Let that of j4 to .ff surest the 
power of altering any magnitude in that ratio." (It is true chat it is not yet 
proved that, B being any magnitude, and /" and Q two magnitudes of the 
same kind, there does exist a magnitude ji which is to JS in the same ratio 
OS /* to Q. It is not till vi. 1 3 that this is proved, by construction, in the 
particular case where the three magnitudes are straight lines. The proof in the 
Greek text of v, 18 which assumes the truth of the more general proposition 
is, by reason of that assumption, open to objection ; see the note on that 
proposition.) Now "every alteration of a magnitude is alteration in some 
ratio, two or more successive alterations are jointly equivalent to but one, and 
the ratio of the initial magnitude to the terminal one is as properly said to be 
the compound ratio of alteration as 13 to be the compound addend in* lieu of 
8 and 5, or 28 the compound multiple for 7 and 4. Competition is used 
here, as elsewhere, for the process of detecting one single alteration which 
produces the joint effect of two or more. The composition of the ratios of 
P \a R, R \.o S, 7" to 6^ is performed by assuming A, altering it in the first 
ratio into B, altering B in the second ratio into C, and C in the third ratio 
into D. The joint effect turns A into D, and the ratio o( A to D is the 
compounded ratia" 

Another word for (ompouncUd ratio is crvn^jK^wot (cruraTmu) which '\% 
common in Archimedes and later writers. 

It is clear that diiplicate ratio, triplicate ratio etc. defined in v. Deff. 9 
and 10 are merely particular cases of compound ratio, being in fact the 
ratios compounded of two, three etc tqual ratios. The use which the Greek 
geometers made of compounded, duplicate, triplicate ratios etc. is well 
illustrated by the discovery of Hippocrates that the problem of the duplication 
of the cube (or, more generally, the construction of a cube which shall be to 
a given cube in any given ratio) reduces to that of finding "two mean 
proportionals in continued proportion." This amounted to seeing that, if 
X, y are two mean proportionals in continued proportion between any two 
lines a, b, in other words, if a is to jc as a; to^, and a: is toj* as jf to #, then a 
cube with side a is to a cube with side .x as a is to b\ and this is equivalent 
to saying that a has to b the triplicate ratio of a to .-t*. 

Euclid is careful to use the forms SorXatri'iuK, tpntXaaiioy, etc. to express what 
we translate as dupiicait, tripliatte etc. ratios ; the Greek mathematicians, 
however, commonly used StirXoffios Xoyg^, "double ratio," TpiirAao-ios Xoyw, 
"triple ratio " etc, in the sense of the ratios of i to i, 3 to i etc. The effort, 
if such it was, to keep the one form for the one signification and the other for 
the other was only partially successful, as there are several instances of the 
contrary use, e,g. in Archimedes, Nicomachus and Pappus. 

The expression for having the ratio which is " duplicate (triplicate) of that 
which it has to the second" is curious — S«rAacrio™ (Tfjin-Aao-i'oi'a) \irfov Ixtw 
^rcp 7rp« TO ieuTt^r — -^Trtp being used as if SiwXafftova or TpiTrXtwtoi'fl were a 
sort of comparative, in the same way as it is used after ftfifom or tAno-o-ovo. 
Another way of expressing the same thing is to say XiJyiK Sn-Xatrtwi' {^fsatXasrlani) 
TO if, %v lyy.... the ratio "duplicate of that (ratio) which,,." The explanation 
of both constructions would seem to be that StirXoo-io; or S(7rAacrw»' is, as 
Hultsch translates it in his edition of Pappus (cf p. 59, 17), duplo maior, 
where the ablative duplo implies not a difference but a proportion. 

The four magnitudes in Def, 10 must of course be in continued proportion 
(Kara to <rv>'(;(n). The Greek text as it stands does not state this. 



t^ o; . BOOK V "^'-nf [v. DEFF, II— 14 

v'..-.;«iai^.r'- m DEFINITION II. "' ''•■" 

'0/M\oya fuyi^Ti KiyiTot To fihi TJytyv/ittva Toij tf)N)ti/i(ViH9 ra Si jira;ic)« Tott 

It is difficult to expre^ the meaning of the Greek in as few words. A 
translation more literal, but conveying less, would be, "Antecedents are called 
wrrtsponding magnitudes to antecedents, and consequents to consequents." 

I have preferred to translate i^Xoyo* by " corresponding " rather than by 
" homologous." I do not agree with Max Simon when he says (Euclid, p. 1 11) 
that the technical term "homologous" is not the adjective i/uSAnyo!, and does 
not mean "corresponding," "agreeing," but "like inrespect of the proportion" 
("ahnlich in Bezug auf das Verhaltniss"), The definition seems to me to be 
for the purpose of appropriating to a technical use precisely the ordinary 
adjective A/ioAoyos, "agreeing" or "corresponding." 

Atttt(tdtnts, TJyov/JLtva, are literally "leoiiing (terms)," and cotistijuents , 
iini/itvo, "following (terms)." 

Definition 12. 

"EvaAAoi Aifyo* foTt Xiji/fii toB jj-you/atrou irpos To -tfytti^tyvi/ mi tou hm^vow 
irpoc TO hm^vGV^ 

We now come to a number of expressions for the transformation of ratios 
or proportions. The first is ifoAAiif, alUrnaldy, which would be better 
described with reference to a proportion of four terms than with reference to 
a ratio. Bui probably Euclid defined all the terms in DeiT. li — 16 with 
reference to ratios because to define them with reference to proportions would 
look like assuming what ought to be proved, namely the legitimacy of the 
various transformations of proportions (cf. v, 16, 7 Por., 18, 17, 19 For,). The 
word iraXXof is of course a common term which has no exclusive reference to 
mathematics. But this same use of it with reference to proportions already 
occurs in Aristotle: Anal. post. t. 5, 74 a 18, i«u *o dvoAcryoi' ort ^™AAi»f, 
"and that a proportion (is true) alternately, or alttrnando" Used with Aoyos, 
as here, the adverb ivaXka^ has the sense of an adjective, "alternate"; we 
have already had it similarly used of " alternate angles " (at JvoAAof yuiruu) in 
the theory of parallels. 

Definition 13. 

AvamAu' Aayot itrrX A^^ii tou iiro/io'ou wc vyov/nrwiu irpo? to ^ytmiitvm/ wc 
Jro/MVOf. 

'AvdiraXiv, " inversely," " the other way about," is also a general term with 
no exclusive reference to mathematics. For this use of it with reference to 
proportion cf, Aristotle, De Cado i. 6, i73b32TiJi' ovaAoyiV ^v to jSiipTj Ix", 
oE xpwoi avatrakai tfoiwiv, " the proportion which the weights have, the times 
will have invtrsely." As here used with Aoyo?, avdvaXw is, exceptionally, 
adjectival. 

Definition 14. 

"Xwvtfri^ Aoyou i<n\ Xi^^4.f TOtJ ^jyavfmivov fttri row iTrofj^fvov <i>s if os wpo* auTo 
TO trontfav. 

The tomfosition of a ratio is to be distinguished from the cetnpounding of 
ratios and compounded ratio {<rvyttiijifvo% Aoyoi) as explained above in the note 



». DKTF. 14— 16] NOTES ON DEFINITIONS ii— 16 133 

on Deff. 9, ro. The fact is that oTPrntftj/it and what serves for the passive of 
it (ffvymtfww) are used for adding as well as compounding in the sense of 
compounding ratios. In order to distinguish the two senses, I have always 
used the word wmpomndo where the sense is that of this definition, though 
this requires a slight departure from the literal rendering of some passages. 
Thus the enunciation of v. 17 says, literally, "if magnitudes compounded be 
in proportion they will also be in proportion separated" (Wv crvyitttfMfa 
/jwytAj ixiktrjov ^, Kol Sto(p«6fl/To omAoyov fo-rai). This practically means 
that, if j4 + 5 is to .S as C + Z) is to A then v4 is to J as C is to D. 
I have accordingly translated as follows : " if magnitudes be proportional 
eompomnde, they will also be proportional separando" (It will be observed 
that stparattdo, a term explained in the next note, is here used, not relatively 
to the proportion /(! is to .ff as C is to D, but relatively to the proportion 
compenmdo, viz. A -^ B \& xo B as, C + Z* is to 2).) The corresponding 
term for eomponendo in the Greek mathematicians is avvBivn, literally "to one 
who has compounded," i.e. " if we compound." (For this absolute use of the 
dative of the participle cf Nicomachus i. 8, 9 otto /ioi'<iSijt...™ia tov SurXaa-ioc 
Xdyoi' jrpojfiupouvTi ni)(pK Aimpow, hoi ictu &v yiviiivTat, ovrot Travrn &pTWiKK 
apTioi tiinv. A very good instance from Aristotle is £(A. Ni(. 1. 5, 1097 b la 
itttt^T^ivovTK yap ^4 roii^ yokcis koX tous diroyovovi koX rw f^ikattf roix ^lXau^ 
tK arfifiov wfxifunv.) A variation for mivSivri, found in Archimedes is nara 
<rvi/$nTLv, Perhaps the more exclusive use of the form <rur0ivTi by geometers 
later than Euclid to denote the composition of a ratio, as compared with 
Euclid's more general use of mvStaxa and other parts of the verb truirtftjfu 
or ovymifuu, may point to a desire to get rid of ambiguity of terms and to 
make the terminology of geometry more exact. 

Definition 15. 

Ataip«r(s Xoyov itm. Xifi^i^ r^v inrt^^i, ^ vvt^pi-^L to i^you^cpoi^ tov 
kico^vvfOy TTpb^ (XvTO rh hrofAivov, 

As composition of a ratio means the transformation, e.g., of the ratio of 
A Xa B into the ratio oi A + B to S, so the uparcUion of a ratio indicates 
the transformation of it into the ratio oi A~ B to B. Thus, as the new 
antecedent is in one case got by adding the original antecedent to the original 
consequent, so the antecedent in the other case is obtained by subtracting the 
original consequent from the original antecedent (it being assumed that the 
latter is greater than the former). Hence the literal translations of tiaifnirn 
Xoyou, "division of a ratio," and of htXovti (the corresponding term to 
<rvr6ivri) as dividtndo, scarcely give a sufficiently obvious explanation of the 
meaning. Heiberg accordingly translates by "subtractio lationis," which 
again may be thought to depart too far from the Greek. Perhaps "separation" 
and separando may serve as a compromise. 



■ *.ir%-»*r!\ jnj 



Definition 16. 

"AKotTTpo^ Ao'yoii /<rTt AiJ^is roG ijyoufiA'Oii nrpos njc vrtpoxijV, ^ ■Airtpiyn 

Conversion of a ratio means taking, e.g., instead of the ratio of A to B, 
the ratio oi A to A —JB {A being again supposed greater than B). As 
iwwTpo^i7 is used for conversion, so ivaxTtpiiftavri is used for conver tends 
(corresponding to the terms avii$ivtt, and SwAdvrt). 



I36 ■■ BOOK V '• ' [v. DEfF. 17, 18 

Definition 17. 

trivia Xafi^avQfiivttfr nal iv T(3 avrcp Aoyitij orav ^ ii^ iy rol^ wp^-rotv fuytOttrt to 
vpwTOr itpm TO itr}(aToy, outuie ivTott Sivripoit fiTytSiiTno irfKuTOk' *poi to JtrjjaToi' ■ 
^ oAXuT' \^>frK T)a¥ axpuiv koB' vw4fyitpt<n.r Tmv jiiam>. 

Si tcrov, tx atquali, must apparently mean ex atquali dislaniia, at an equal 
distance or interval, i.e. after an equal number of intervening terms. The 
wording of the definition suggests that it is rather a proportion ex atquali 
than a ratio ex aet/uali which is being defined (cf. Def. 12). The meaning is 
clear enough. If a, 6,c,d...ht one set of magnitudes, and A, B, C, D... 
another set of magnitudes, such that 

(I is to * as vi is to B, 

^ is to f as .d is to C, 
and so on, the last proportion being, e.g., 

/' is to /, as A* is to Z, 
then the inference ex aeqtmli is that 

a is to / as /4 is to i. ' 

The/orf that this is so, or the truth of the inference from the hypothesis, 
is not proved until v. 22. The definition is therefore merely verbal; it gives 
a convenient name to a certain inference which is of constant application in 
mathematics. But ex aequali could not be intelligibly defined except with 
reference to two sets of ratios respectively equal. , 



Definition 18. 

TtTopayjiivTi Si A.vaXoyia ttniv^ orav t^loiv ^VTtffV fMytS^ ircu aXAai»' aurotc 
r<r«i' TO Trk^ftov yiVip-ai ut /xiv iv to« jfpcoToit fHyiSiirtv^oviityoir irpos Irofitvov, 
ovT4i>c iy Tots otvTtpot.^ fiiyi&tiTtv ^tivfitvtty wpo^ Iwofi-tvov, mi Si iv rots Tr/xJroi^ 
lktyiBi<rKV inop^fvoy irpos aXka Tc, ourcii? iy this £<UTCpot; aXXo Tt vphi ^yovfttvov^ 

Though the words Bi' 'urav, ex atqiitUt, are not in this definition, it gives a 
description of a case in which the inference ex aeguali is still true, as will be 
hereafter proved in v. 23, A perturbed proportion is an expression for the 
case when, therp being three magnitudes a, b, c and three others A, B, C, 

a is to # as .^ is to C, 

and ^ is to f as j4 is to if. 

Another description of this case is found in Archimedes, "the ratios being 
dissimilarly ordered " (nVo^oitut TtTayji-iyiay ruy koymy). The full description of 
the in/ereiue in this case (as proved in v. 23), namely that 

a is to f as .f4 is to C, 

is ex aequali in perturbed proportion (8c' lO-ou iv Tviapayp,ivj} dvaXoyl^), 
Archimedes sometimes omits the &' Eo-ou, first giving the two profwrtions and 
proceeding thus: "therefore, the proportions being dissimilarly ordered, a has 
to e the same ratio as A has to C." 

The fact that Def. 18 describes a particular case in which the inference 
St' to-oii will be proved true seems to have suggested to some one after 
Theon's time the interpolation of another definition between 17 and 18 eo 



v.DEF. »8] NOTES ON DEFINITIONS 17, 18 137 

describe the ordinary case where the argument ex atquaii holds good. The 
interpolated definLtion runs thus ; "an ordered proportion (T<ray/it»T; aVoXoyw) 
arises when, as antecedent is to consequent, so is antecedent to consequent, 
and, as consequent is to something else, so is consequent to something else." 
This case needed no description after Def, 17 itself j and the supposed 
definition is never used. 

After the definitions of Book v. Simson supplies the following axioms. 

I. Equimultiples of the same or of equal magnitudes are equal to one 
another. 

3. Those magnitudes of which the same or equal magnitudes are 
equimultiples are equal to one another. 

3. A multiple of a greater magnitude is greater than the same multiple 
of a less. 

4. That magnitude of which a multiple is greater than the same multiple 
of another is greater than that other magnitude. 



.I". . r." .1:. T 



i. . ..; 



jL»u'.."fi 1' 

l" - t.-.rli ,^>, i 



I .1 

\ 



I i, 



I- ' • <>• 



BOOK V, PROPOSITIONS. 



Proposition 



If there be any number of magnitudes ■whatever which are, 
respectively, equimultiples of any magnitudes equal in multitude, 
then, whatever multiple one of the magnitudes is of one, that 
multiple also will all be of all. 

Let any number of magnitudes whatever AB, CD be 
respectively equimultiples of any magnitudes E, F equal in 
multitude ; 

I say that, whatever multiple AB is of E, that multiple will 
AB, CD also be of E, F. 



For, since ^^ is the same multiple of E that CD is of F, 
as many magnitudes as there are in AB equal to E, so many 
also are there in CD equal to F. 

Let AB be divided into the magnitudes AG, GB equal 
to E, 

and CD into CH, HD equal to F ; 

then the multitude of the magnitudes AG, GB will be equal 

to the multitude of the magnitudes CH, HD. 

Now, since ^6^ is equal to E, and CH to F, 
therefore AG is equal to E, and AG, CH to E, F. 

For the same reason 

GB is equal to E, and GB, HD to E, E; 

therefore, as many magnitudes as there are in AB equal to £, 
so many also are there in AB, CD equal to E, F; 



V. 1, a] PROPOSITIONS i, a 139 

therefore, whatever multiple AB Is of E, that multiple will 
AB, CD also be of E, F. 
Therefore etc. 

Q. E, n. 

De Morgan remarks of v, i — 6 that they are "simple propositions of 
concrete arithmetic, covered in language which makes them iminte)!igible to 
modem ears. The lirst, for instance, states no more than that len acres and 
Un roods make ten times as much as one acre and one rood." One aim 
therefore of notes on these as well as the other propc^itions of Book v. 
should be to make their purport clearer to the learner by setting them side by 
side with the same truths expressed in the much shorter and more familiar 
modem (algebraical) notation. In doing so, we shall express magnitudes by 
the first letters of the alphabet, a, b^ c etc., adopting small instead of capital 
letters so as to avoid confusion with Euclid's lettering ; and we shall use the 
small letters ot, n,p etc to represent integral numbers. Thus ma will always 
mean m times a or the m"' multiple of a (counting i . a as the first, i . a as the 
second multiple, and so on). 

Prop. I then asserts that, if ma, mb, mc etc. be any equimultiples of a, b, i 
etc, then 

ma*mb-^mt+ ...=m (a + i + c+...). 

Proposition 2, 

J/ a first magnitude be the same multiple of a second 
that a third is of a fourth, and a fifth also be the same multiple 
of the second thai a sixth is of the fourth, the sum of the first 
and fifth will also be the same multiple of the second that the 
sum of the third and sixth is of the fourth. 

Let a first magnitude, AB, be the same multiple of a 
second, C, that a third, DE, 
is of a fourth, F, and let a ^ , , b g 



fifth, EG, also be the same 

multiple of the second, C, that ° ^ ^ 

a sixth, EH, is of the fourth °" ' ' — ' — ' ' ' 

F; F 

I say that the sum of the 

first and fifth, AG, will be the same multiple of the second, C, 

that the sum of the third and sixth, DN, is of the fourth, F, 

For, since AB is the same multiple of C that DE is of F, 
therefore, as many magnitudes as there are in AB equal to C, 
so many also are there in D£ equal to F. 

For the same reason also, 
as many as there are in BG equal to C, so many are there 
also in £// equal to F; 



»4» ■ , BOOK V [v. a, 3 

therefore, as maay as there are in the whole AG equal to C, 
so many also are there in the whole DN equal to F. 

Therefore, whatever multiple AG is of C, that multiple 
also is DM of F. 

Therefore the sum of the first and fifth, AG, is the same 
multiple of the second, C, that the sum of the third and sixth, 
DH, is of the fourth, F. . , 

Therefore etc. 

Q, E, D. 

To find the corresponding formula for the result of this proposition, we 
may suppose rt to be the " second " magnitude and b the " fourth." If now 
the " first " magnitude is ma, the " third " is, by hypothesis, mb ; and, if the 
"fifth " magnitude is na, the "si)cth" is nf>. The proposition then asserts that 
ma + na is the same multiple of a that mh-^ nb'\s of *. 

More generally, if /a, ya... and j>i, gfi... be any further equimultiples of 
a, b respectively, ma + na-^-pa-yqa-^ ■■■ is the same multiple of a that mb-y 
fib+pb -yqb -y- ,., ts of b. This extension is stated in Simson's corollary to 
V. a thus ; 

" From this it is plain that, if any number of magnitudes AB, BG., GH 
be multiples of another C; and as many DE, EK, KL be the same 
multiples of F, each of each ; the whole of the first, viz. AH, is the same 
multiple of C that the whole of the last, viz. DL, is of F" 

The course of the proof, which separates m into its units and also n into 
its units, practically tells us that the multiple of a arrived at by adding the 
two multiples is the {« + n)th multiple ; or practically we are shown that 

i»«n- «(j = {w 4 n) a, 
or, more generally, that 

ma + /la +pa + ... ={»i + 11 +/ + , , ,) o. 



- " . Proposition 3, ■ ..»>•< , 

J/ a first magnitude be the same multiple of a second 
that a third is of a fourth, and if equimultiples be taken 
of the first and third, then also ex aequali the magnitudes 
taken will be equimultiples respectively, the one of the second 
and the other of the fourth. 

Let a first magnitude A be the same multiple of a second 
B that a third C is of a fourth D, and let equimultiples MF, 
GH be taken oi A, C; 
I say that £F is the same multiple of S that G/f is of D. 

For, since £F is the same multiple of A that G// is of C, 
therefore, as many magnitudes as there are in £F equal to 
A, so many also are there in G// equal to C. 



T. $] PROPOSITIONS 2, 3 t4t 

Let £F be divided into the magnitudes £JC, KF equal 
to A, and GH mXo the magnitudes GL, LH equal to C\ 

then the multitude of the magnitudes EK, ^/^will be equal 
to the multitude of the magnitudes GL, LH. 



A- 
B- 

E- 
C- 
D - 
O- 



And, since A is the same multiple of B that C is of /), 

while EK is equal to A, and GL to C 

therefore EK is the same multiple of B that GZ, is of D. 
For the same reason ,' 

KF is the same multiple of B that LH is of Z?. 

Since, then, a first magnitude EK is the same multiple 
of a second B that a third GL is of a fourth D, 
and a fifth KF is also the same multiple of the second B that 
a sixth LH is of the fourth D, 

therefore the sum of the first and fifth, EF, is also the same 
multiple of the second B that the sum of the third and sixth, 
GH, is of the fourth Z>. [v. 2] 

Therefore etc, 

Q, E. D, 

Heiberg remarks of the use of ex aeqiiali in the enunciation of this projK}- 
sition that, strictly speaking, it has no reference to the definition {17) of a 
ratio fx atquaU. But the uses of the expression here and in the definition 
are, I think, sufficiently parallel, as may be seen thus. The proposition 
asserts that, if 

na, nb are equimultiples of a, b, 
and if m .na, m . nh are equimultiples of na, nb, 

then M , na is the same multiple of a that m .nils of *. Clearly the proposi- 
tion can be extended by taking further equimultiples of the last equimultiples 
and so on ; and we can prove that 

p .f...t!t.nais the same multiple o( a that/ ,q...m.nb is of ^, 
where the series of numbers p .q...m .n is exactly the same in both 
expression^ ; 

and tx atquali (&' laov) expresses the fact that the equimultiples are at the 
same dhtanee from 3, # in the series na, m .na... and nb, m.nb... respectively. 



^100 ^ , BOOK V 1 ; [v. 3, 4 

Here again the proof breaks m into its units, and then breaks n into its 
units ; and we are ptacticalty shown that the multiple of a arrived at, viz. 
m . rta, is the multiptt: denoted by the product of the numbers m, h, Le. the 
(m«)th multiple, or in other words that 



Proposition 4. ' 

// a first magnitude have to a second the same ratio as a 
third to a fourth, any equimultiples whatever of the first and 
third will also have the sanu ratio to any equimultiples 
whatever of the second and fourth respectively, taken in 
corresponding order. 

For let a first magnitude A have to a second B the same 
ratio as a third C to a fourth D \ and let equimultiples E, F 
be taken of A, C, and G, H other, chance, equimultiples of 
B,D\ 
I say that, as E is to G, so is Flo H. , 

A ■ ■ ■•' 

B 

E ' ■ 

• Q . ■ I 

K 1 

M- 1 1 

C— — - 
D- 

F 1 

■ . ... L 1 

N . 1 



For let equimultiples A', Z. be taken of E, F, and other, 
chance, equimultiples M, JV of G, H. 

Since E is the same multiple of A that F is of C, 

and equimultiples K, L oi E, ./^have been taken, 

therefore K is the same multiple of A that L is of C, [v. 3] 

For the same reason 

J/ is the same multiple of B that A'' is of Z?, 



T. 4] PROPOSITIONS 3, 4 143 

" ' And, since, as A is to B,so\% C to D, • • ' - • ' 
and oi A, C equimultiples K, L have been taken, 
and of ^, D other, chance, equimultiples M, N, 
therefore, if K is in excess of M, L also is in excess of N^ 
if it is equa], equal, and if less, less, [v. Def, 5] 

And K, L are equimultiples of E, F, 
and My TV other, chance, equimultiples of G, H \ 
therefore, as £" is to G, so is F to H. [v. Def. 5] 

Therefore etc. 

Q. E. D. 
This proposition shows that, if a, b, (, d are proportionals, then 
«ki is to n^ as Mf is to nd; w. -t^ : 

and the proof is as follows : 

Take pma, pmc any equimultiples of ma, mc, and qnb, qnd any equimulti- 
ples of «*, nd. 

Since a : i=^ : rf, it follows [v. Def. 5] that, 

according as pma ■> = < gnb, pmc-> = < qnd. 
But the^ and ^-equimultiples are any equimultiples; , 

therefore [v. Def. 5] . „ , - 

ma : nd = me : nd. 

It will be observed that Euclid's phrase for taking any equimultiples of 
A, C and any other equimultiples of .5, Z> is " let there be taken equimulti- 
ples E, F o{ A, C, and G, H other, chance, equimultiples of B, D," E, F 
being called \aaXK woUaTAoirui simply, and G, H o^Aa, a \^v•)^tv, EiriutK 
iroAAairAao-ia. And similarly, when any equimultiples (/T, L) of E, F 
come to be taken, and any other equimultiples {M, N) of G, H. But 
later on Euclid uses the same phrases about the nrtu equimultiples with 
reference to the original magnitudes, reciting that " there have been taken, of 
A^ C, equimultiples K, L and of B, D, other, chamt, equimultiples M, JV" ; 
whereas M, JV are not any equimultiples whatever of B, D, but are any 
equimultiples o( the parlicu/ar multiples {G, //) which have been taken of £, 
D respectively, though these tatter have been taken at random. Simson would, 
in the first place, add 5 trvytv in the passages where any equimultiples E, F 
are taken of A, C and any equimultiples A', £ are taken of E, F, because the 
words are "wholly necessary" and, in the second place, would leave them 
out where M, iVare called oAAn, a inxty, liraKii TrokJuurKaria of B, D, because 
it is not true that of B^ D have been taken "any equimultiples whatever (« 
hvyt), M, N." Simson adds: "And it is strange that neither Mr Bri^s, who 
did right to leave out these words in one place of Prop. 13 of this book, nor 
Dr Gregory, who changed them into the word ' some ' in three places, and 
left them out in a fourth of that same Prop, 13, did not also leave them out 
in this place of Prop. 4 and in the second of the two places where they occur 
in Prop. 1 7 of this book, in neither of which they can stand consistent with 
truth : And in none of all these places, even in those which they corrected in 
their Latin translation, have they cancelled the words « hvx'^ '" 'he Greek 
text, as they ought to have done. The same words S. iToj(i are found in 



144 BOOK V [v. 4 

four places of Prop. 1 1 or this book, in the first and last of which they are 
necessary, but in the second and third, though they are true, they are quite 
superfluous ; as they likewise are in the second of the two places in which 
they are found in the 12th prop, and in the like places of Prop. 21, 33 of this 
book; but are wanting in the last place of Prop. 13, as also in Prop, 25, 
Book XI," 

As will be seen, Sirason's emendations amount to alterations of the text 
so considerable as to suggest doubt whether we should be justifled in making 
them in the absence of MS, authority. The phrase " equimultiples of A, C 
and other, chance, equimultiples of £, D " recurs so constantly as to suggest 
that it was for Euclid a quasi-stereotyped phrase, and that it is equally genuine 
wherever it occurs. Is it then absolutely necessary to insert i trvxt in places 
where it does not occur, and to leave it out in the places where Simson holds 
it to be wrong ? I think the text can be defended as it stands. In the first 
place to say "take equimultiples of A, C" is 3. fair enough way of saying 
take any equimultiples whatevtr of A, C. The other difliculty is greater, but 
may, I think, be only due to the adoption of any whatever as the translation 
of a, Xrv-jif. As a matter of fact, the words only mean chance equimultiples, 
equimultiples which are the result of random selection. Is it not justifiable 
to describie the product of two chance numbers, numbers selected at random, 
as being a " ekance number," since it is the result of two random selections ? 
1 think so, and I have translated <i cruxc accordingly as implying, in the case 
in question, " other equimultiples whatever they may happen to be," 

To this proposition Theon added the following : 

" Since then it was proved that, if K is in excess of M, L is also in excess 
of N, if it is equal, (the other is) equal, and if less, less, 
it is dear also that, 

if ^ is in excess of A!*, A'' is also in excess of Z, if it is equal, {the other is) 
equal, and if less, less ; 
and foi this reason, 

as C is to £, so also is HXa F. 

PosiSM. From this it is manifest that, if four magnitudes be proportional, 
they will also be proportional inversely." 

Simson rightly pointed out that the demonstration of what Theon intended 
to prove, viz. that, if E, G, F, H be proportionals, they are proportional 
inversely, i.e. (7 is to .£ as /T is to /^ does not in Che least depend upon this 
4th proposition or the proof of it ; for, when it is said that, " if A" exceeds M, 
Z also exceeds N etc.," this is not proved from the fact that E, G, F, H are 
proportionals (which is the conclusion of Prop. 4), but from the fact that 
A, B, C, D are proportionals. 

The proposition that, if A, B, C, D are proportionals, they are also 
proportionals inversely is not given by Euclid, but Simson supplies the proof 
in his Prop. B. The fact is really obvious at once from the 5th definition 
of Book V. (cf, p. 127 above), and Euclid probably omitted the proposition 
as unnecessary. 

Simson added, in place of Theon's corollary, the following : 

" Likewise, if the first has the same ratio to the second which the third 
has to the fourth, then also any equimultiples whatever of the first and third 
have the same ratio to the second and fourth : And, in like manner, the first 
and the third have the same ratio to any equimultiples whatever of the second 
and fourth," 



V. 4. 5] PROPOSITIONS 4, 5 r^ 

The proof, of course, Tollows exactly the method of Euclid's proposition 
itself, with the only difference that, instead of one of the two pairs of equi- 
multiples, the magnitudes themselves are taken. In other words, the conclu- 
sion that 

MM is to n^ as KM' is to m^ i •■■•i 

is equally true *hen either #« or « is equal to unity. 

As De Morgan says, Simson's corollary is only necessary to those who will 
not admit jl/'into the list M, lAf, 3 J/' etc.; the exclusion is grammatical and 
nothing else. The same may be said of Simson's Prop. A to the effect that, 
" If the first of four magnitudes has to the secotid the same ratio which the 
third has to the fourth : then, if the first be greater than the second, the third 
is also greater than the fourth ; and if equal, equal ; if less, less." This is 
needless to those who believe ona A to be a proper component of the list of 
multiples, in spite of mullut signifying many. 



Proposition 5. ' 

1/ a magnitude be the same multiple of a magnitude that 
a part subtracted is of a part subtracted, the remainder will 
also be the same multiple of the remainder that the whole is of 
the whole. 

S For let the magnitude AB ht. the same multiple of the 
magnitude CD that the part AE subtracted is of the part CF 
subtracted ; 

I say that the remainder EB is also the same multiple of the 
remainder ED that the whole AB is of the whole CD. 

: . , . ^— 1 1 ? 1 1 ? 



10 For, whatever multiple AE is uf CF, let EB be made 
that multiple of CG. 

Then, since AE is the same multiple of CF that EB ts 
of GC, 
therefore AE is the same multiple of CF that AB is of GF, 

[V. .] 

IS But, by the assumption, AE is the same multiple of CF 
that AB is of CD. 

Therefore AB is the same multiple of each of the magni- 
tudes GF, CD ; 

therefore GF is equal to CD. 
» Let CF be subtracted from each ; 
therefore the remainder GC is equal to the remainder FD. 



I4« 



BOOK V 



[»-S 



And, since AE is the same multiple of CF that EB is of 
GC, 

and GC is equal to DF, 
as therefore AE is the same multiple of CF that ^5 is of FD. 
But, by hypothesis, 

AE is the same multiple of CF that -(4^ is of CD ; 
therefore EB is the same multiple of FD that -^^ is of CZ?, 
That is, the remainder EB will be the same multiple of 
30 the remainder FD that the whole AB is of the whole CD. 
Therefore etc. 

Q, E. D, 

10. let EB be made that muJtiple of CO, ntavn-rKinaf ytYmirw rol to EB tw 
rB. From this way of stating; the construction one mi^t suppose that CG was given and 
EB had to be found equal to a certain multiple of it. But in fact EB ia what is given and 
CG has to be found, i.e. CG has to be constructed as a certain ju^uttiplc of EB* 



This proposition correspotids to V. i, with subtraction taking the place of 
addition. It proves the foimula 

ma~mb = M{a—b). '"" ' 

Euclid's construction assumes that, ii AE\% any multiple of CF^ and EB 
is any other magnitude, a fourth straight line can be found such that EB is 
the same multiple of it that AE is of CF, or in other words that, given any 
magnitude, we can divide it into any number of equal parts. This is however 
not proved, even of straight lines, much less other magnitudes, until vi. 9. 
Peletarius had already seen this objection to the construction. The difficulty 
is not got over by regarding it merely as a hyfothetkal construction ; for 
hypothetical constructions are not in Euclid's manner. The remedy is to 
substitute the alternative construction given by Sim son, after Peletarius and 
Campanus' translation from the Arabic, which only requires us to add a 
magnitude to itself a certain number of times. The demonstration follows 
Euclid's line exactly. 

"Take AG the same multiple of FD that AE is of CF; 

therefore AE is the same multiple of C^that EG is of CD. 

But AE, by hypothesis, is the same multiple of CF that 
.^^ is of CD ; therefore EG is the same multtple of CD that 
A£ is of CD; 

wherefore ^C is equal to AJ3. 

Take from them the common magnitude AE ; the remainder 
AG is equal to the remainder EB. 

Wherefore, since AE is the same multiple of CFthu AG is 
of FD, and since AG is eqaa.1 to EB, 
therefore AE is the same multiple of CFtitat EB is o^ FD, 

But AE is the same multiple of C^that AB is of CD; 
therefore EB is the same multiple of FD that AB is of CD." 



V. 5, 6] PROPOSITIONS 5, 6 147 

Euclid's proof amounts to thb. 

Suppose a magnitude x taken such that ■• ■' 

ma — mi^mx, say. 
Add TTii to each side, whence (by v. i) 

Therefore a=jf + #, or *=a — *, ' *' ' ' 

so that ■ mi-tni> = m{a-b). ' ' 

Simson's proof, on the Other hand, argues thus. 

Take x = m{a~ i), the same multiple of (a — i) that md is ol d. 

Then, by addition of mfi to both sides, we have [v. i] 
x + m6 = ma, 
or at = ma — mb. • • 

That ia, ma~mi = m{a — i). 

Proposition 6, 

If (wo magnitudes be equimultiples 0/ two magnitudes, and 
any magnitudes subtracted from them 6e equimultiples of the 
sam4, the remainders also are either equal to the same or equi- 
multiples of them. 

For let two magnitudes AB, CD be equimultiples of two 
magnitudes E, F, and let AG, CH 

subtracted from them be equi- a q 8 

multiples of the same two E, F; ' ' ' 

I say that the remainders also, GB, ^ o h 

HD, are either equal to E, F or — *■ — 1 — >- — 1 — 
equimultiples of them. F — 

For, first, let GB be equal to ^ ; 
I say that HD is also equal to F. 

For let CK be made equal to F. 

Since AG\% the same multiple of E that CH is of F, 
while GB is equal to E and KC to F, 
therefore AB is the same multiple of E that KH is of F. 

[v. 2] 

But, by hypothesis, AB is the same multiple of E that 
CDv^QiF; 
therefore KH is the same multiple of F that CD is of F. 

Since then each of the magnitudes KH, CD is the same 
multiple of F, 

therefore KH ts equal to CD, 



148 ^ BOOK V [v. 6, 1 

Let C// be subtracted from each ; 
therefore the remainder A'C is equal to the remainder //D. 

But F is equal to JCC ; 
therefore //D is also equal to J^. 

Hence, if GB is equal to £, HD is also equal to F. 

Similarly we can prove that, even if GB be a multiple 
of B, HD is also the same multiple of F. 

Therefore etc 

Q. E, D. 

This proposition corresponds to v. 3, with subtraction taking the place of 
addition. It asserts namel)' that, if n \& less than m, ma — na is the same 
multiple of a that mb-nb'\% of b. The enunciation distinguishes the cases in 
which m-» is equal to i and greater than i respectively. 

Simson observes that, while only the first case ^the simpler one) is proived 
in the Greek, both are given in the Latin translation from the Arabic ; and 
he supplies accordingly the proof of the second case, which Euclid leaves to 
the reader. The fact is that it is exactly the same as the other except that, in 
the construction, CK is made the same multiple of /"that GB is of E, and 
at the end, when it has been proved that KC is equal to HD, instead of 
concluding that'/fZ> is equal to F, we have to say " Because GB is the same 
multiple of E that KC is of F, and KC is equal to HD, therefore HD is 
the same multiple of ./^that GB is of E." 

Proposition 7, i. .. - > 

Equal magnitudes have to (he same the same ratw, as also 
has the same io equal magnitudes. 

Let A, B be equal magnitudes and C any other, chance, 
magnitude ; 

I say that each of the magnitudes A, B has the same ratio 
to C, and C has the same ratio to each of the magnitudes 
A,B. 



A D^ 

B Er. 



Ci < f- 



For let equimultiples D, E o^ A, B be taken, and of C 
another, chance, multiple F. 

Then, since D is the same multiple of A that E is of B, 
while A is equal to ^, 

therefore D is equal to E. 

But F is another, chance, magnitude. 



V. 7. 8] PROPOSITIONS 6—8 149 

If therefore D is in excess of F, E is also in excess of F, 
if equal to it, equal ; and, if Jess, less. 

And D, E are equimultiples oi A, B, 
while F'ls another, chance, multiple of C; 

therefore, as A is to C, so is B to C, [v. Def. 5] 

I say next that C also has the same ratio to each of the 
magnitudes A, B. 

For, with the same construction, we can prove similarly 
that D is equal X,o E\ 

and F is some other magnitude. 

If therefore F\%\n excess of D, it is also in excess of E, 
if equal, equal ; and, if less, less. 

And /^ is a multiple of C, while D, E are other, chance, 
equimultiples of ^, B ; 

therefore, as C is to -r^ , so is C to B. [v. Def. 5] 

Therefore etc. 

PoRiSM, From this it is manifest that, if any magnitudes 
are proportional, they will also be proportional inversely. 

Q. E. D. 

In this proposition there is a similar use of t irvx^y to that which has 
been discussed under Prop. 4. Any multiple F <A C is taken and then, 
four lines lower down, we are told that " F is another, chance, magnitude." 
It is of course not any magnitude whatever, and Simson leaves out the 
sentence, but this time without calling attention to it. 

Of the Porism to this proposition Heiberg says that it is properly put here 
in the best ms.j for, as August had already observed, if it was in its right 
place where Theon put it (at the end of v. 4), the second part of the proof of 
this proposition would be unnecessary. But the truth is that the Porism is no 
more in place here. The most that the proposition proves is that, if A, B 
are equal, and Cany other magnitude, then two conclusions are simultaneously 
established, (1) that A is to C s.% B is to Cand (2) that C\%io A &.& C is to 
B. The second conclusion is not established from the first conclusion (as 
it ought to be in order to justify the inference in the Porism), but from a 
hypothesis on which the first conclusion itself depends ; and moreover it is 
not a proportion in its genera! form, i.e. between four magnitudes, that is in 
question, but only the particular case in which the consequents are equal. 

Aristotle tacitly assumes inversion (combined with the solution of the 
problem of Eucl. vi. 11) in Meteoroiogiea ni. 5, 37G a 14 — 16. 

Proposition 8. 

Of unequal magnitudes, the greater has to the same a 
greater ratio than the /ess has ; and the same has to the less 
a greater ratio than it has to the greater. 







'. ^ t» * ■< 







h 






- 


l-T..' 






L 




, ,. 1 








N 







tgp 9- BOOK V fr.» 

-" Let AB, C be unequal magnitudes, and let AB be greater ; 

let D be another, chance, - n 

magnitude ; 

r say that AB has to Z? a 

greater ratio than C has to 

£), and /? has to C a greater 

ratio than it has to AB. 

For, since ^^ is greater 
than C, let BB be made equal 
toC; 

then the less of the magni- 
tudes A£, JSB, if multiphed, 
will sometime be greater than £f. [v, Def. 4] 

ICase I.] 

First, let A£ be less than BB ; 

let AB he multiplied, and let BG be a multiple of it which is 
greater than D ; 

then, whatever multiple BG is of AB, let G/f be made the 
same multiple of BB and ^ of C ; 

and let Z be taken double of D, M triple of it, and successive 
multiples increasing by one, until what is taken is a multiple 
of D and the first that is greater than K, Let it be taken, 
and let it be N which is quadruple of D and the first 
multiple of it that is greater than a. 

Then, since K is less than N first, 

therefore K is not less than M. 

And, since BG is the same multiple of AB that G/f is of 
BB, 

therefore BG is the same multiple of AB that B/f is of AB. 

[V. ,] 

But BG is the same multiple of AB that A' is of C ; 

therefore F/f is the same multiple of AB that A' is of C ; 

therefore B//, K are equimultiples of AB, C. 

Again, since GH is the same multiple of EB that K is 
of C 

and EB is equal to C, 

therefore GH is equal to A'. 



i/^ti PROPOSITION 8 f$i 

But K is not less than M; ■ -^ s ■ 

therefore neither is (7/^ less than j)/. .1 .• • • 1 

And FG is greater than Z? ; 
therefore the whole J^ff is greater than Z>, M together. 

But D, M together are equal to A', inasmuch as M is 
triple of D, and M, D together are quadruple of D, while 
A'' is also quadruple of D ; whence M, D together are equal 
\o N. 

But FH is greater than M, D ; 

therefore /^^ is in excess of A^, ■ "-•-»"• 

while K is not in excess of N. ' 

And FH, K are equimultiples of AB, C, while N is 
another, chance, multiple of D ; 

therefore AB has to D a. greater ratio than C has to D. 

[v. Def. 7] 

I say next, that D also has to C a greater ratio than D 
has to AB. 

For, with the same construction, we can prove similarly 
that TV is in excess of K, while N is not in excess of FH, 

And A^ is a multiple of/?, 
while FH, K are other, chance, equimultiples of AB, C ; 

therefore D has to C a greater ratio than D has to AB. 

[v, Def. 7] 

\Case 2.] 

Again, let AE be greater than EB. 

Then the less, EB, if multiplied, will sometime be greater 
than D, \y. Def. 4] 

Let it be multiplied, and e q 

let GH be a multiple of EB * ' 

and greater than D ; ^ ' ^ ^ 

and, whatever multiple GH is ' ' ' ' 

of EB, let FG be made the "^ ^ — ' 

same multiple of AE, and K ^ ' _' • •■ 

of C L ■ ^ ..^ 

Then we can prove simi- " ' ' ' 

larly that FH, K are equi- "^ ' ' ' • 

multiples of AB, C\ 

and, similarly, let N be taken a multiple of D but the first 

that is greater than FG, 

so that FG is again not less than M. 



IS> BOOK V [v. 8 

But GH is greater than D ; 
therefore the whole FH is in excess of D, M, that is, of N. 

Now A' is not in excess of N', inasmuch as FG also, which 
is greater than GH, that is, than K, is not in excess of N. 

And in the same manner, by following the above argu- 
ment, we complete the demonstration. 

Therefore etc. 

Q. E. I). 

The two separate cases found in the Greek text of the demonstration can 
practically be compressed into one. Also the expositor of the two cases 
makes them differ more than they need. It is necessary in each case to 
select the smaller of the two segments AE, EB of AB with a view to taking 
a multiple of it which is greater than D ; in the first case therefore A£ is 
taken, in the second EB. But, while in the first case successive multiples of 
D are taken in order to find the first multiple that is greater than (7^ (or A"), 
In the second case the multiple is taken which is the first that is greater than 
EG. This difference is not necessary; the first multiple of Z) that is greater 
than G/f would equally serve in the second case. Lastly, the use of the 
magnitude /C might have been dispensed with in both cases ; it is of no 
practical use and only lengthens the proofs. For these reasons Simson 
considers that Theon, or some other unskilful editor, has vitiated the 
proposition. This however seems an unsafe assumption ; for, while it was 
not the habit of the great C J reek geometers to discuss separately a number of 
different cases (eg, in i, 7 and t. 35 Euclid proves one case and leaves the 
others to the reader), there are many exceptions to prove the rule, e.g. Eucl, 
III. 15 and 33 ; and we know that many fundamental propositions, after- 
wards proved generally, were first discovered in relation to particular cases 
and then generalised, so that Book v., presenting a comparatively new 
theory, might fairly be expected to exhibit more instances than the earlier 
books do of unnecessary subdivision. The use of the JC is no more con- 
clusive against the genuineness of the proofs. 

Nevertheless Simson 's version of the proof Is certaimy snorter, and more- 
over it takes account of the case in which AE is efua/ to £B, and of the case 
in which AE, EB are both greater than D (though these cases are scarcely 
worth separate mention). 

" If the magnitude which is not the greater of the two AE, EB be (i) 
not less than D, take FG, G/f the doubles of AE, EB. 

But if that which is not the greater of the two AE, EB be (2) less than 
£>, this magnitude can be multiplied so as to become greater than £> whether 
it he AE or EB, 

Let it be multiplied until it becomes greater than D, and let the other be 
multiplied as often ; let EG be the multiple thus taken of AE and GJ/ the 
same multiple of EB , 
therefore EG and G/f are each of them greater than D, 

And, in every one of the cases, take Z the double of D, M its triple and 
so on, till the multiple of Z) be that which first becomes greater than GH. 

Let N be that multiple of D which is first greater than ff/^ and j^the 
multiple of D which is next less than N. 



V. 8, 9] PROPOSITIONS 8, 9 133' 

Then, because iV is the multiple of J) which is the first that becomes 
greater than GJf, ■ ■ ■:. . 

the next preceding tnultipU is not greater than G/f; 
that is, Gff is not less than M. 

And, since FG is the same multiple of AE that G/f is of EB, 
GH'is ttie same multiple of EB that FH moi AB\ [v. 1] 

wherefore FH, C/f are equimultiples of AB, EB. 

And it was shown that Glf^as not less than Af; 

and, by the construction, FG is greater than D\ 
therefore the whole FH\s greater than M, D together. ' , 

But M, D together are etjual to N ; 
therefore FH\% greater than N. , 

But Gff'ss not greater than N; 
and FH, GHa-tn equimultiples of AB, BE, 

and jVis a multiple of D\ 
therefore AB has to /? a greater ratio than BE (or C) has to D. [v. Def, 7] 

Also D has to BE a greater ratio than it has to A/i. 

For, having made the same construction, it may be shown, in like manner, 
that N is greater than GH but that it is not greater than FH; 
and TV is a multiple of D, 1 .^ ; . 

and GH, FH zxa equimultiples of EB, AB; 

Therefore Dhas,to EB a greater ratio than it has to AB." [v, Def. 7] 

The proof may perhaps be more readily grasped in the more symbolical 
form thus. 

Take the w;th equimultiples of C, and of the excess of AB over C (that is, 
oi AE), such that each is greater than D\ 

and, of the multiples of/?, let ^i? be the first that is greater than mC, and nD 
the ne<;t less multiple of D. 

Then, since wC is not less than nD, • • * ' 

and, by the construction, m{AE) is greater than D, 

the sum of wCand rii(AE) is greater than the sum of aD and D. 

That is, m(AB) is greater than/Z>. ... 

And, by the construction, mC is less than pD. 

Therefore [v. Def. j] AB has to D a. greater ratio than C has to D. 

Again, since //J is less than m(AB), 

i.nApD is greater than mC, <'- ■ < •' "> ' 

D has to C a greater ratio than D has to AB. ' ' '' ■■'■■' 

• *' '. 

Proposition 9. 

Magnitudes which have the same ratio to the same are 
equal to one another ,■ and magnitudes to which the same has 
the same ratio are equal. 



tU BOOK V [v. 9 

For let each of the magnitudes ^, B have the same 
ratio to C ; 
I say that A is equal to B. 

For, otherwise, each of the 
magnitudes A, B would not ° 

have had the same ratio to C\ [v. 8] 

but it has ; 

therefore A is equal to B. 

Again, let C have the same ratio to each of the magni- 
tudes A, B ; 
I say that A is equal to B. 

F'or, otherwise, C would not have had the same ratio to 
each of the magnitudes A, B \ \y-^\ 

but it has ; 

therefore A is equal to B. 

Therefore etc. 

Q. E. D. 

If ii^ is to C as B is to C, 
or if C is to V* as C is to B, then A is equal to B. 

Simson gives a more expticit proof of this proposition which has the 
advantage of referring back to the fundamental sth and 7th definitions, 
instead of quoting the results of previous projxjsitions, which, as will be seen 
from the next note, may be, in the circumstances, unsafe. 

" Let A, B have each of them the same ratio to C\ 

A is equal to B. 

For, if they are not equal, one of them is greater than the other ; 
let A be the greater. 

Then, by what was shown in the preceding proposition, there are some 
equimultiples of A and B, and some multiple of C, such that the multiple of 
A is greater than the multiple of C, but the multiple of B is not greater than 
that of C. 

Let such multiples be taken, and let ZJ, j£ be the equimultiples of A, B, 
and F the multiple of C, so that D may be greater than F, and E not greater 
than F. 

But, because vf is to C as ^ is to C, 
and of A, B are taken equimultiples D, £, and of C is taken a multiple F, 
and I> is greater than F, 

E must also be greater than F. [v. Def. s] 

But £ is not greater than F: which is impossible. 

Next, let C have the same ratio to each of the magnitudes A and S ; 
A is equal to B, 

For, if not, one of them is greater than the other ; 
let A be the greater. 



V. 9, lo] PROPOSITIONS 9, 10 iJS 

Therefore, as was shown in Prop. 8, there is some multiple F of C, and 
some equimultiples E and D al B and A, such that F is greater than E and 
not greater than D. 

But, because C is to -^ as C is to A, 
and /"the multiple of the first is greater than E the multiple of the second, 

^the multiple of the third is greater than D the multiple of the fourth. 

[v, Def, 5] 

But ^ is not greater than D : which is impossible. 

Therefore A is equal to B." 



, , , ^ Proposition id. ,.. .^ ,, 

0/ magnitudes which have a ratio to (fie same, that 
which has a greater ratio is greater ; and that to which the 
same has a greater ratio is less. 

For let A have to C a greater ratio than B has to C ; 
I say that A is greater than B. 



For, if not, A is either equal to B or less, 

Now /4 is not equal to B\ 
for in that case each of the magnitudes A, B would have 
had the same ratio to C ; [v. 7] 

but they have not ; 

therefore A is not equal to B. 

Nor again is A less than B ; 
for in that case A would have had to C a less ratio than B 
has to C ; [v. 8] 

but it has not ; 

therefore A is not less than B. 

But it was proved not to be equal either ; 
therefore A is greater than B. 

Again, let C have to ^ a greater ratio than C has to A ; 
I say that B is less than A. 

For, if not, it is either equal or greater. ' ' 

Now B is not equal to A ; 
for in that case C would have had the same ratio to each of 
the magnitudes A, B ; [v. 7] 

but it has not ; 

therefore A is not equal to B. 



tSU BOOK V [v. lo 

Nor again is ^ greater than W ; ' ' *■ • -> > 

for in that case C would have had to ^ a less ratio than it 
has to A ; [v. 8] 

but it has not ; ' •' •'= 

therefore B is not greater than A. 
But it was proved that it is not equal either ; 

therefore B is less than A. 
Therefore etc. Q. E. D- 

No better example can, I think, be found of the acuteness which Simson 
brought to bear in his critical examination of the £&mcnfs, and of his great 
services to the study of Euclid, than is furnished by the admirable note on 
this proposition where he points out a serious flaw in the proof as given in 
the text. 

For the Rrst time Euclid is arguing about greater and ieis ratios, and it 
will be found by an examination of the steps of the proof that he assumes 
more with regard to the meaning of the terms than he is entitled to assume, 
having regard to the fact that the definition of greater ratio (Def, 7) is all 
that, as yet, he has to go upon. That we cannot argue, at present, about 
greaifr and less as applied to mtwi m the same way as about the same terms 
in relation to nrngniludes is indeed sufficiently indicated by the fact that Euclid 
does not assume for ratios what is in Book i. an axiom, viz. th;it things which 
are equal to the same thing are equal to one another ; on the contrary, he 
proves, in Prop. 11, that ratios which are the same with the same ratio are the 
same with one another. 

Let us now examine the steps of the proof in the text. First we are told 
that 

"j4 is greater than B. 

For, if not, it is either equal to B or less than it. 

Now jJ is not equal to B ; 

for in that case each of the two magnitudes A, B would have had the 
same ratio to C: [v. 7] 

but they have not : 

therefore A is not equal to B" 

As Simson remarks, the force of this reasoning is as follows. 

If A has to C the same ratio as B has to C, 
then — ^supposing any equimultiples of A, B to be taken and any multiple 
of C— 

by Def. 5, if the multiple of /i be greater than the multiple of C, the multiple 
of B is also greater than that of C. 

But it follows from the hypothesis (that ^ has a greater ratio to C than B 
has to C) that, 

by Def. 7, there must be some equimultiples of A, B anA somt multiple of 
C such that the multiple of ^ is greater than the multiple of C, but the 
multiple of B is not greater than the same multiple of C. 

And this directly contradicts the preceding deduction from the supposition 
that A has to C the same ratio as B has to C ; 

therefore that supposition is impossible. 



V. lo] PROPOSITION lo ij*f 

The proof now goes on thus : 

*' Nor again is A less than B ; 
for, in (hat case, A would have had to C a less ratio than B has to C\ 

'^ but tt has not ; 

therefore A is not less than B." 

It is here that the difficulty arises. As before, we must use Def. 7. "A 
would have had to C a less ratio than B has to C," or the equivalent state- 
ment that B would have had to C a greater ratio than A has to C, means 
that there would have been same equimultiples of B, A and some multiple of 
C such that 

(i) the multiple of B k greater than the multiple of C, but 

(z) the multiple of .1^ is nat greater than the multiple of C, 
and it ought to have been proved that this can never happen if the hypothesis 
of the proposition is true, vh. that A has to C a greater ratio than B has to 
C: that is, it should have been proved that, in the latter case, the multiple of 
A is always greater than the multiple of C whenever the multiple of B is 
greater than the multiple of C (for, when this is demonstrated, it will be 
evident that B cannot have a greater ratio to C than A has to C). But this 
is not proved (cf. the remark of De Morgan quoted in the note on v, Def 7, 
p. 130), and hence it is not proved that the above inference from the supi>osi- 
tion that A is less than B is inconsistent with the hypothesis in the enunciation. 
The proof therefore fails. 

Sim son suggests that the proof is not Euclid's, but the work of some one 
who apparently "has been deceived in applying what is manifest, when 
understood of magnitudes, unto ratios, viz. that a magnitude cannot be both 
greater and less than another," 

The proof substituted by Simson is satisfactory and simple. . , i , r 

"Let A have to Ca greater ratio than B has to C; 
A is greater than B. 

For, because A has a greater ratio to C than B has to C, there are some 
equimultiples of A, B and some multiple of C such that 

the multiple of A is greater than the multiple of C, but the multiple of B 
is not greater than it. [v. Def. 7] 

Let them be taken, and let D, E \y& equimultiples of A, B, and F a 
multiple of C, such that 

^ , D\% greater than F^ 

bilt ' £ is not greater than F. « 

Therefore D is greater than E. 

And, because D and E are equimultiples of A and B, and D is greater 
than E, 

therefore A is greater than B. [Simson's 4th Ax.] 

Next, let C have a greater ratio to B than it has \a A\ 
B is less than A. 

For there is some multiple F of C and some equimultiples E and D ai B 
and A such that 

^is greater than E but not greater than D. [v. Def. 7] 

Therefore E is less than D ; 
and, because E and D are equimultiples of B and A, :■ -i -m ' 

therefore B is less than A." 



IS* BOOK V [v. II 

,. . _ -t; 

M . _ . Proposition i i. " " 

Ratios which are the same with the same ratio are also 
the same with one another. 

For, as W is to B, so let C be to jD, 
and, as C is to D, so let ^ be to A; ' 

I say that, as A is to B, so \% E Xo F. , .'i 



A- 
B- 

o- 

L- 



c 


E 


D 


F' 




,, 






M 


N- 



For of Ay C, E let equimultiples G, H, K be taken, and 
oi B, D, Mother, chance, equimultiples L, M, N. 

Then since, as ^4 is to B, so is C to D, 
and of y4, C equimultiples G, /Thave been taken, 
and of B, D other, chance, equimultiples L, M, 
therefore, if C^ is in excess of Z, H \s also in excess of M, 
if equal, equal, 
and if less, less. 

Again, since, as C is to D, so is E to E, 
and of C, E equimultiples H, K have been taken, 
and of D, F other, chance, equimultiples M, N, 
therefore, if H is in excess of M, K is also in excess of N, 
if equal, equal, 
and if less, less. •. ^ ■ ^ -i 

But we saw that, if H was in excess of M, G was also 
in excess of Z- ; if equal, equal ; and if less, less ; 
so that, in addition, if G is in excess of Z, K is also in excess 
kAN, 

if equal, equal, „ , , ,. ,., „ ,, . ,^,^ ^ . 

and if less, less. 

And G, K are equimultiples oi A, £, 
while L, N are other, chance, equimultiples of B, F; 

therefore, as A is to B, so is E to E. -i 

Therefore etc, ' 



V. II, i»] PROPOSITIONS II, I » i$9 

Algebraically, if - a -.b-c: d, ' .« 

atid c\d=t\f, 

then a:b = e\f. 

The idiomatic use of the imperfect in quoting a result previously obtained 
is noteworthy. Instead of saying " But it was proved that, if H is in excess 
of M, G is also in excess of L," the Greek text has "But if H was in excess 
of M, G was also in excess of L," oXAa tl vittfulyft to © rov M, hvipaxt noi 

TO H ToZ A. 

This proposition is tacitly used in combination with V. i6 and v. 14 in the 
geometrical passage in Aristotle, Miteorohgica 111. 5, 376 a a 2 — *6, j 



Proposition 12. 

If any number of magnitudes be proportional, as one of 

the antecedents is to one of the consequents, so will all the 
antecedents be to all the consequents. 

Let any number of magnitudes A, B, C, D, E, F be 
proportional, so that, as A is to B, so is C xo D and E 
to F\ 
I say that, as A is to B, so are A, C, E to B, D, F. 



fi. 


B- 


— 


C 





e 


- 


F 



















M- 




K 







For o{ A, C, E let equimultiples G, N, K be taken, 
and q{ B, D, i^ other, chance, equimultiples L, M, N. 

Then since, as A is to B, so is C to D, and E to F, 
and of A, C, E equimultiples G, H, K have been taken, 
and of B, D, F other, chance, equimultiples L, M, N, 
therefore, if G is in excess of Z, /^ is also in excess of M, 
and^ofiV, 
if equal, equal, 
and if less, less ; 
so that, in addition, 

if G is in excess of L, then G^ H, K are in excess of Z,, M, N, 
if equal, equal, 
and if less, less, • i ' - ^ 



i6o ,. BOOK V [v. 13, 13 

Now G and G, H, K are equimultiples of A and A, C, E, 
since, if any number of magnitudes whatever are respec- 
tively equimultiples of any magnitudes equal in multitude, 
whatever multiple one of the magnitudes is of one, that 
multiple also will all be of all. [v. i] 

For the same reason 

L and L, M, N are also equimultiples of B and B, D, F\ 

therefore, as A is to B, so are A, C, E lo B, D, P. 

[v. Def. s] 
Therefore etc. 

Q. E. D. 

Algebraically, \i a : a' = b : b' = e : / eic, each ratio is equal lo the ratio 
{a + 6 + e+ ...) : (a' +^'4-^+ ...). 

This theorem is quoted hy Aristotle, El A, Nk. v. 7, 1 131 b 14, in the 
shortened form "the whole is to the whole what each part is to each part 
(respectively)." 

Proposition 13. 

If a first magnitude have to a second the same ratio as a 
third to a fourth, and the third have to the fourik a greater 
ratio than a fifth has to a sixth, the first will also have to the 
second a greater ratio than the fifth to (he sixth. 

For let a first magnitude A have to a second B the 
same ratio as a third C has to a fourth D, 

and let the third C have to the fourth D a greater ratio than 
a fifth E has to a sixth E; 

I say that the first A will also have to the second B a greater 
ratio than the fifth E to the sixth E. 



f, o ■ M- 

B — D N 



E- 
F- 

L- 



For, since there are some equimultiples of C, E, 

and of D, E other, chance, equimultiples, such that the 
multiple of C is in excess of the multiple of D, 



V. 13) PROPOSITIONS ii, 13 xBi 

while the multiple of £ is not in excess of the multiple of P, 

[v. Def.. 7] 
let them be taken, 

and let G, Hhe. equimultiples of C, E, 

and K, L other, chance, equimultiples of D, F, 

so that G is in excess of K, but H is not in excess of L ; 

and, whatever multiple G is of C, let M be also that multiple 

of ^, 

and, whatever multiple K is of D, let N be also that multiple 

of^. 

Now, since, as ^ is to B, so is C to D, 
and Qi A, C equimultiples M, G have been taken, 
and of B, D other, chance, equimultiples A^, K, 
therefore, if M is in excess of N, G is also in excess of K, 
if equal, equal, 
and if less, less. , .. , . ;, li' [v. Def. 5] 

But 6^ is in excess of AT ; .' 1. c 

therefore M is also in excess of A^. ' ' 

But H is not in excess of Z ; ' • 

and M, /^are equimultiples of ^, E, ., 

and jV, L other, chance, equimultiples of ^, F\ 

therefore A has to 5 a greater ratio than E has to F. 

[v. Def. j] 
Therefore etc. 

Q. E. D. 

Algebraically, if a\h = t\dt ~* ■' • 

and e : d->e :/ , ' , 

then a \ b-rt\f. 

After the words " for, since "in the first line of the proof, 'Hieon added 
" C has to i> a greater ratio than E has to F" so that " there are some 
equimultiples" b^an, with him, the principal sentence. 

The Greek text has^ after " of D, F other, chance, equimultiples," " and 
the multiple of C is in excess of the multiple of D...." The meaning being 
" such that," I have substituted this for " and," after Simson. 

The following will show the method of Euclid's proof. 

Since e:d->e\f, 

there will be some equirnultiples me, nu of t, e, and some equimultiples nd, ^ 
oftf,/, auch that 

momt, while me'^nf. 



i«« BOOK V [v. 13, 14 

But, since a:i = c;d, (. ,:/v * •) 

therefore, according as ma > = <ni, mc> — < nd. 

And mi>»d; 
therefore ma -> nb, while (from above) mel^nf. 
Therefore a\b>t\f. 

Simson adds as a corollary the following : 

" If the first Kave a greater ratio to the second than the third has to the 
fourth, but the third the same ratio to the fourth which the fifth has to the 
sixth, it may be demonstrated in like manner that the first has a greater ratio 
to the second than the fifth has to the sixth." 

This however scarcely seems to be worth separate statement, since it only 
amounts to changing the order of the two parts of the hypothesis. 

Proposition 14. 

If a first magnitttde have to a second the same ratio as a 
third has to a fourth, and the first be greater than the third, 
the second will also he greater than the fourth; if equal, equal; 
and if less, less. 

For let a first magnitude A have the same ratio to a 
second .^ as a third C has to a fourth D\ and let A be 
greater than C ; 
I say that B Is also greater than D. 

A c 

8 D 



For, since A is greater than C, 
and B is another, chance, magnitude, 
therefore A has to .5 a greater ratio than C has to B. [v, 8] 

But, as ^ is to B, so is C to Z* ; 
therefore C has also to /? a greater ratio than C has to B. 

[v. Jl] 

But that to which the same has a greater ratio is less ; 

[v. 10] 
therefore D is less than B ; 
so that B is greater than D. 

Similarly we can prove that, if .^4 be equal to C, B will 
also be equal to D ; 

and, if A be less than C, B will also be less than D. 
Therefore etc. 



T. 14, is] propositions 13—15 163 

Algebraically, if a : i = t : d, 

then, according aso> = <f, fi> = <<f, 
Simson adds the specific proof of the second and third parts of this 
proposition, which Euclid dismisses with "Similarly we can prove...." 

" Secondly, if v^ be equal to C, B is equal to D\ for /4 is to 5 is C, that 
is A, is to Z> ; 

therefore B is equal to D. [v. 9] 

Thirdly, if ^4 be less than C, B shall be less than D, 
For C is greater than A ; 
and, because C'\iXo D »s A is to B, 

D is greater than B, by the first case. >, , . 

Wherefore B is less than D." 

Aristotle, Mtteorol. iii. 5, 376 a ti— i4i quotes the equivalent proposition 
that, if a>^, e->d. 



Proposition 15. 

Parts have the same ratio as the same multiples of them 
taken in corresponding order. 

For let AB be the same multiple of C that DE is of /"; 
I say that, as C is to F, so\s AB to DE. 



Af 1 1 tB Ct- 

Oi ■ ' 'E f'- 



For, since AB is the same multiple of C that DE is of E, 
as many magnitudes as there are in AB equal to C, so many 
are there also in DE equal to E. 

Let AB be divided into the magnitudes AG, GH, HB 
equal to C, 

and DE into the magnitudes DK, KL, LE equal to E \ 
then the multitude of the magnitudes AG, GH, HBW\\\ be 
equal to the multitude of the magnitudes DK, KL, LE. 

And, since AG GH, HB are equal to one another, 
and DK, KL, LE are also equal to one another, 
therefore, as AG is to DK, so is GH to KL, and HB to LE. 

{^- 7] 

Therefore, as one of the antecedents is to one of the 
consequents, so will all the antecedents be to all the 
consequents ; [v. la] 

therefore, as ^G is to DK, so is AB to DE. 



i«4 -- BOOK V •"• [v. IS, '6 

But AG is equal to C and DK to F; 

therefore, as C is to i^ so is AS to I^S, 
Therefore etc. v,, , q, e. d. 

Algebraically, a : b~ma : mi. 

Proposition i6. 

If four magnitudes be proportional, they will also be 
proportional alternately. 

Let A, B, C, D )x. four proportional magnitudes, 
so that, as A is to ^, so is C to Z? ; 

I say that they will also be so alternately, that is, as W is 
to C, so is B to D. 

A c 



o- 



E< 1 1 1 Qi 1 

Fi 1 1^ 1 Hi '-—I 

For o{ A, B let equimultiples E, F be taken, ,, ,i . 
and of C, D other, chance, equimultiples G, H. 

Then, since E is the same multiple of A that F is of B, 
and parts have the same ratio as the same multiples of 
them, [v. is] 

therefore, as ^ is to ^, so is £* to F. 

But as ^ is to 5, so is C to /? ; 
therefore also, as C is to D, so is E to F. [v. ii] 

Ag^in, since G, H are equimultiples of C, D, 
therefore, as C is to /?, so is C to H. [v. ij] 

But, as C is to Z?, so is ^ to F\ 
therefore also, as ^ is to F, so is G to H. [v. n] 

But, if four magnitudes be proportional, and the first be 
greater than the third, 

the second will also be greater than the fourth ; 
if equal, equal ; 
and if less, less. •. ,-v-y^,- [v- m] 

Therefore, if E Is in excess of G, F is also in excess of H, 
if equal, equal, 
and if less, less. 



V. i6] PROPOSITIONS 15, i6 1C5 

Now E, 7^ are equimultiples oi A, B, 
and G, H other, chance, equimultiples o{ C, D\ 

therefore, as A is to C, so is B to D. [v. Def. s] 

Therefore etc. 

Q, E. D. 

3, " Let A, B, C, D be four proportignsl magnitudes, so that, as A Is to B, so fs 

C to D." In a number of expressions like this it is absolutely necefisffty, wheit translatitig 
inta Endbh, to interpolate words which are not in the Greek. Thus the Greek here is ; 
litfTup Tiaaapa liiP^&Ti d^^Xtyyoi' r^ A, U, T, A, i^t rh A rp6% TO B, oih-iifj ti r rpii tA A, 
literally *' Let At B^ C, D he four proportional magnitudes, as ^ to ^, 60 C to Z>^" The 
same remark applies to the eotresponding expressions in the neil proposiiions, v. 17, 18, 
and to other forms of expression in V. to — 13 and later propositions : e^g. in v, 10 we have 
a phrase meaning literally '*I^t there be mngnitudes... which taken two and two are in the 
same ratio, as if to ^, so Z> to £," etc.: in v. it " (magnitudes)... which taken two and 
two are in the same r^tio, atvd ht the proportion of them be perturbed, as ^ to ^, so 
£ to /",'' etc. In all such cases (where the Greek is so terse as to be almost ungrammatical) 
I shall insert the words necessary in English, without further remark. 

Algebraically, if a : b^c : d, 

then a; ( = b : d. 

Taking equimultiples /fta, mb of a, b, and equimultiples m, nd of (,d, we 
have, by v. 15, 

a : b = Ma : mb, 

c ; d=ne : nd. 

And, since a : b = e ; d, 

we have [v. 11] ma : mb = n£ md. 

Therefore [v. 14], according as wa > = < «<r, mb> = <ii4,' 

so that a : e = b : d. 

Aristotle tacitly uses the theorem in MetetfrologUa ill. 5, 376 a xz — 34. 
The four magnitudes in this proposition mtist all be ^ the same kind, and 
Simson inserts " of the same kind " in the enunciation. 

This is the first of the propositions of Eucl. v. which Smith and Bryant 
{Euclid's Ekmtnti of Geomttry, tgoi, pp. 298 sqq.) prove by means of vi, i 
so far as the only geometrical magnitudes in question are straight lines or 
rectilineal areas \ and certainly the proofs are more easy to follow than 
Euclid's. The proof of this proposition is as follows. 

To prove that, If Jour magnitudes of the same kind [straight lines or 
rectilineal areas] be proportionals, they will be proportionals -when taken 
alttmately. 

Let F, Q, /{, Sbe the four magnitudes of the same kind such that 

P:Q = R:S; 

then it is required to prove that '. ■^ . 1 < ; 

P-.R^Q-.S. \ -' 

First, let all the magnitudes be areas. 

Construct a rectangle abed equal to the area P, and to be apply the 
rectangle beef equal to Q, 

Also to cd>, bf apply rectangles ag, bk equal to JF, S respectively. ' ' ■ 



m 



BOOK V 



[v. i6, ij 



Then, since the rectangles ac, be have the same height, they are to one 
another as their bases, [vl. i] 

Hence P:Q = ah:bf. 

But P:Q = R:S. 

Therefore R •S = ab:b/y [v. 1 1] 

i.e. rect. ag : rect. At = ab : bf. 

Hence (by the converse of vi. i) the rect- 
angles ag, bk have the same height, so that k 
is on the line kg. 

Hence the rectangles ae, ag have the same 
height, namely ab ; also ^, bk have the same 
height, namely h/. 

Therefore rect. ac • rect ag=bc \ 

and rei;t be : recL bk = bc ; bg. 

Therefore rect, ac : rect, ag - recL be ; rect, bk. 

That is, P:Ji=Q:S. 

Se(ondl}\ let the magnitudes be straight lines AB, BC, CD, DE. 
Construct the rectangles Ab, Be, Cd, Dt with the same height. 



i t t 


a 






f 


b 






h 3 


k 



bg. 



[VI- 
[v. Il] 



a 


be d e 










A f 


i i 


J 1 


3 E 



Then Ab\Bc = AB\BC, 

aiid Cd : De= CD : DE. 

But AB:BC=CD:DE. 

Therefore Ab .Bc=Cd: Dt. 

Hence, by the first case, 

Ab: Cd=Be-De, 
and, since these rectangles have the same height, 
AB: CD = BC : DE. 



[VI. i] 
[V. „] 



Proposition 17. 

1/ magnitudes be proportional componendo, they will also 
be proportional separando. 

Let AB, BE, CD, DF be magnitudes proportional com- 
ponendo, so that, as AB is to BE, so is CD to DF\ 
I say that they will also be proportional separando, that is, 
as AE is to EB, so is CF to DF. 

For of AE, EB, CF, FD let equimultiples GH, HK, 
LM, MN be taken, 
and of EB, FD other, chance, equimultiples, KO, NP, 



V. 17] PROPOSITIONS i6, ij t6? 

Then, since GH is the same multiple of AE that HK is 
oiEB, 

therefore GH is the same multiple of AE that GK is of AB. 

[V. ,] 

But GH is the same multiple of AE that LM is of CF\ 
therefore GK is the same multiple of AB that LM is of C/". 



"E — B e — r~B 



H K O 



Again, since LM is the same multiple of CF that MN 
is of FD, 

therefore LM is the same multiple of CF that LN is of CD. 

Iv. l] 

But LM was the same multiple of CF that CA' is of AB \ 

therefore GK is the same multiple of AB that LN is of CD. 

Therefore GK^ LN 3.r& equimultiples oi AB, CD. 

Again, since HK is the same multiple of EB that MN is 
of/Z?, 

and KO is also the same multiple of EB that NP is of /"/>, 
therefore the sum HO is also the same multiple of EB that 
MP is of /^£'. [v. i] 

And, since, as AB is to ^^, so is CD to /?/% 

and oi AB, CD equimultiples GK, LN have been taken, 

and of EB, FD equimultiples HO, MP, 

therefore, if GK is in excess of HO, LN is also in excess of 
MP, 

if equal, equal, , . i, 

and if less, less. 

Let GK be in excess of HO ; 

then, if HK be subtracted from each, ' 

GH is also in excess of KO. 

But we saw that, if GK was in excess of HO, LN was 
also in excess of MP ; 

therefore LN is also in excess of MP, i 



ag^ BOOK V [v. 17 

and, if MN be subtracted from each, 

LM is also in excess of NP ; 
so that, if GH is in excess of KO, LM is also in excess of 
NP. 

Similarly we can prove that, 
if GH be equal to KO, LM will also be equal to NP^ 
and if less, less. 

And GH, LM are equimultiples of AE, CF, 

while KO, NP are other, chance, equimultiples of EB, FD ; 

therefore, as AE is to EB, so is CF to FD. 

Therefore etc. 

Q. E. D. 

Algebraically, if a \b = c ; d, 

then {a-b):b = (c-d)\d. 

I have already noted the somewhat strange use of the participles of 
(TuyKttcr^cu and Stai/xur^at to convey the sense of the technical ^^ivB^a^^ and 
WiptiTK Xoyou, or what we denote by (ompontndo and sefarafido. lax 
<rvyi«iV«'a fiiyiOr) dirdkoyov J, itai StaifitOivra draXjryov IcrtOi is, literally, "if 
magnitudes compounded be proportional, they will also be proportional 
separated," by which is meant "if one magnitude made up of two parts is to 
one of its parts as another magnitude made up of two parts is to one of its 
parts, the remainder of the first whole is to the part of it first taken as the 
remainder of the second whole is to the part of it first taken." In the 
algebraical formula above a, c are the wholes and b,a-b and d, c-^are the 
parts and remainders respectively. The formula might also be stated thus ; 

If a-^b •.b=-c ^d vd, 

then a : b = c : d, 

in which case a + #, c + d are the wholes and 0, a and d, i the parts and 
remainders respectively. Looking at the last formula, we observe that 
"separated," Siatp«fl(fTa, is used with reference not to the magnitudes a, i, c, d 
but to the comfamndid magnitudes a + b, b, c + d, d. 

As the proof is somewhat long, it will be useful to give a conspectus of it 
in the more symbolical form. To avoid minuses, we will takf for the 
hypothesis the form 

a + b\s xa b zs c + d\sUi d. 

Take any equimultiples of the four magnitudes a, b, t, d, viz. 

ma, mb, ntt, md, 
and any other equimultiples of the consequents, viz. 

nb and nd. 
Then, by v. i, m{a + b), ni {c + d) are equimultiples of a + i, c-¥d, 
and, by v. 2, (m + «) b, {tn + n)d are equimultiples of b, d. 
Therefore, by Def, s, since a + b is to 6 its c+d is to d, 

according as (w (« + i) > = <{« + /«) #, ot {f + <()>-< (m + «) rf. 



V. I J, i8] PROPOSITIONS t;, i8 z^ 

Subtract from m (a + i), (m + n)i the common part mi, and from 
m(c + d), {m + »)d the common part md; and we Itave, 

according as ma> = <n6, mc> = <nd. 
But ma, mc are any equimultiples of «, e, and nb, nd any equimultiples of 
kd, 

therefore, by v. Def. 5, 

a is to ^ as r is to d. 

Smith and Bryant's proof follows, mutatis mutandis, their alternative proof 
of the next proposition (see pp. 173 — 4 below). 



Proposition 18. 

If magnitudes be proportional separando, they will also be 
proportional com pone ndo. 

Let AE, EB, CF, FD be magnitudes proportional 
separando, so that, as AE is 
to EB, so is CF to FD ; a e b 

I say that they will also be ' ' ^ 

proportional componendo, that a ^ & 

is, as AB is to BE, so is .1 

CD to FD. 

For, if CD be not to DF as AB to BE, 

then, as AB is to .5.5', so will CD be either to some 
magnitude less than DF or to a greater. 

First, let it be in that ratio to a less magnitude DG, 

Then, since, as AB is to BE, so is CD to DG, 

they are magnitudes proportional componendo; 

so that they will also be proportional separando. [v. 17] 

Therefore, as AE is to EB, so is CG to GD. 

But also, by hypothesis, 

as AE is to EB, so is CF to FD. 

Therefore also, as CG is to GD, so is CF to FD. [v, it] 

But the first CG is greater than the third CF; 

therefore the second GD is also greater than the fourth 
FD. [v. 14] 

But it is also less : which is impossible. 
Therefore, as AB is to BE, so is not CD to a less 
m^nitude than FD. 



Ifo BOOK V [v. i8 

Similarly we can prove that neither is it in that ratio to 
a greater; 

it is therefore in that ratio to FD itself. 
Therefore etc. 



Q. E. D, 



Algebraically, if a \b = e : d^ 

then {a^b) \ b^{f-kd) ; d. 



In the enunciation oC this proposition there is the same special use of 
Siijp>jli.(ya and crvtri^iVra as there was of cruyKtiftira and SuupiSiyra in the 
last enunciation. Practically, as the algebraical form shows, Stgpriiiira, might 
have been left out. 

The following ts the method of proof employed by Euclid, 

Given tnat a:b = e:d, 

suppose, if possible, that 

(o + jS) ■.b = {c + d):{d±x). 
Therefore, J<^ir«i«<fo [v. 17], 

a : b = {c + x) •.{d±x), 
whence, by v. 1 1, {cT x) ; {d ±x) = c : d. 

But {e—x)< c, while {d + x)> d, 

and {c-\-x)>e, while {d-x)<d, 

which relations respectively contradict v. 14. 

Simson pointed out (as Saccheri before him .saw) that Euclid's demonstra- 
tion is not legitimate, because it assumes without proof that to any three 
magnitudes, two of which, at ieasi, are 0/ the same kind, there exists a fourth 
pr&poriionai. Clavius and, according to him, other editors made this an 
a;<iom. But it is far from axiomatic ; it is not till vi. 1 2 that Euclid shows, 
by construction, that it is true even in the particular case where the three 
given magnitudes are alt straight lines. 

In order to remove the defect it is necessary either (r) to prove beforehand 
the proposition thus assumed by Euclid or (2) to prove v. tS independently 
of it. 

Saccheri ingeniously proposed that the assumed proposition should be 
proved, for areas and straight Hues, by means of Euclid vi. i, 2 and 12. As 
he says, there was nothing to prevent Euclid from interposing these proposi- 
tions immediately after v. 17 and then proving v. 18 by means of them. 
VI. 12 enables us to construct the fourth proportional when the three given 
magnitudes are straight lines ; and vi. 1 2 depends only on vi. i and 2. 
" Now," says Saccheri, " when we have once found the means of constructing 
a straight line which is a fourth proportional to three given straight lines, we 
obviously have the solution of the general problem ' To construct a straight 
line which shall have to a given straight line the same ratio which two polygons 
have {to one another).'" For it is sufficient to transform the polygons into 
two triangles of equal height and then to construct a straight line which shall 
be a fourth proportional to the bases of the triangles and the given straight 
line. 

The method of Saccheri is, as will be seen, similar to that adopted by 



V. i8] PROPOSITION i8 ift 

Smith and Bryant {/i^. at.) in proving the theorems of Euclid v. i6, 17, 18, 21, 
so far as straight lines and rectilineal areas are concerned, by means of vi. i. 

De Morgan gives a sketch of a general proof of the assumed proposition 
that, B being any magnitude, and P and Q two magnitudes of the same kind, 
there does exist a magnitude A which is to ^ in the same ratio as /" to Q. 

" The right to reason upon any aliquot part of any magnitude is assumed ; 
though, in truth, aliquot parts obtained by continual bisection would suffice : 
and It is taken as previously proved that the tests of greater and of less ratio 
are never both presented in any one scale of relation as compared with 
another" (see note on v. Def, 7 ad ^n,). 

"(i) If ^be to ^ in a greater ratio than Pto Q, so is every magnitude 
greater than AT, and so are s&me leu magnitudes ; and if jW be to ^ in 
a less ratio than P to Q, so is every magnitude less than M, and so are 
some greater magnitudts. Part of this is in every system : the rest is proved 
thus. If j^ be to J in a greater ratio than P to Q, say, for instance, we find 
that isjI/ lies between »i2? and 23i?, while 15/" lies before J2Q. Let \^M 
exceed 22^? by Z; then, if iV be less than M by anything less than the 15th 
part of Z, i^N is between ziB and 23^; or JVJ less than M, is in a greater 
ratio to B than P to Q. And similarly for the other case, 

(2) ,^can certainly be taken so small as to be in a less ratio to B than 
P to Q, and so large as to be in a greater ; and since we can never pass from 
the greater ratio back again to the smaller by increasing M, it follows that, 
while we pass from the first designated value to the second, we come upon an 
intermediate magnitude A such that every smaller is in a less ratio to B than 
P to Q, and every greater in a greater ratio. Now A cannot be in a less ratio 
to B than P to Q, for then some greater magnitudes would also be in a less 
ratio ; nor in a greater ratio, for then some less magnitudes would be in a 
greater ratio; therefore A is in the same ratio to 2? as ^ to Q. The previously 
proved proposition above mentioned shows the three alternatives to be the 
only ones." 

Alternative proofs of V. 18. 

Simson bases his alternative on v, 5, 6, As the 18th proposition is the 
converse of the 17th, and the latter is proved by means of v. i and j, of 
which V. 5 and 6 are converses, the proof of v. 18 by v. 5 and 6 would be 
natural; and Simson holds that Euclid must have proved v, i3 in this way 
because "the sth and 6th do not enter into the demonstration of any 
proposition in this book as we have it, not can they be of any use in any 
proposition of the Elements," and "the sth and 6th have undoubtedly been 
put into the 5th book for the sake of some propositions in it, as all the other 
propositions about equimultiples have been." 

Simson's proof is however, as it seems to me, intolerably long and difficult 
to follow unless it be put in the symbolical form as follows. 

Suppose that a is to i as (^ is to rf; , ..",.'" ; 

it is required to prove that a 4- j is \a b »&c-¥d\&X.o i. 

Take any equimultiples of the last four magnitudes, say 

w(a + i), iiib, m{c + d), md, 

and any equimultiples of i, d, as 

nb, nd. 



^fM. BOOK V [v. iS 

Oearly, if «d is greater than mi, ... 

lid is greater than md; 
if equal, equal ; and if less, less. 
I Suppose nd not greater than mi, so that nd is also not greater than W. 

Now m(a4-i') is greater than mi ; 

therefore m[a + i) is greater than »6. 

Similarly m (^ + rf) is greater than nd. 

II. Suppose ni greater than mi. 

Since « (a + i), md, m{e + d), md are equimultiples of (o + i), i, (f + rf)i <^> 
ma is the same multiple of a that m{a-y i) is of {a + i), 
and mc is the same multiple of c that m (f + d) is of (c + d), 

so that ffio, mc are equimultiples of a, c. [v. 5] 

Again n^, nd are equimultiples of J, d, 
and so are m^, md\ 
therefore (n-m)i, {H-m)d are equimultiples of b, d and, whether n-m 
is equal to unity or to any other integer [v. 6^ it follows, by Def. S, that, 
since a, b, r, d are profxsrtionals, 
if ma is greater than {n-m)i, 

then mc is greater than {n-m)d; 

if equal, equal ; and if less, less. 

(i) If now m{a-\-i} is greater than ni, subtracting mi from each, we have 
ma is greater than {n-m)i; 
therefore mc is greater than (n - m)d, 

and, if we add md to each, 

m(c-¥d) is greater than nd. 

(3) Similarly it may be proved that, 
if w (a + ^) is equal to ni, 

then j» (f + rf) is equal to nd, 

and (3) that, if m(aA- i) is less than tti, 

then mic + d) is less than nd. 

But (under I. above) jt was proved that, in the case where ni is not 
greater than mb, 

m{a +i) a always greater than ni, 

and m(c + d) is always greater than nd. 

Hence, whatever be the values of m and n, m {c + d) is always greater than, 
equal to, or less than nd according as m(a + i) is greater than, equal to, or 
less than nb. 

Therefore, by Def. 5, 

a+i is U>baac + d is tod, 

Todhunter gives the following short demonstration from Austin {Exami- 
nation 0/ the fin t six books of Euclid's Elements). 

"Let AE be to J?J? as CFis to FD: 

AB shall be to BE as CD is to DF. 



V. i8] PROPOSITION i8 ^ 

For, because AE is to EB as CF'xs to FD, 
therefore^ alternately, 

AE is to CFtts EB is to FD. [v. i6] 

And, as one of the antecedents is to its consequent, so is the sum of the 
antecedents to the sum of the consequents: [v. 12] 

therefore, as EB is to FD, so are AE, EB together to CF, 
FD together ; 

that is, AB is to CD as EB is to FD. 

Therefore, alternately, 

AB is to BE as CD is to FD." 

The objection to this proof is that it is only valid in the case 
where the proposition v. t6 used in it is valid, i.e. where all four 
magnitudes aie of the same kind. 

Smith and Bryant's proof avails where all four magnitudes 
are straight lines, where all four magnitudes are rectilineal areas, 
or where one antecedent and its consequent are straight lines and the other 
antecedent and its consequent rectilineal areas. .. , 

Suppose that A : B= C : D. 

First, let all the magnitudes be areas. 

Construct a rectangle abed equal to A, and to be apply the rectangle baf 
equal to B. 

Also to ab, bj apply the rectangles ag, bk 
equal to C, D respectively. 

Then, since the rectangles m, be have equal 
heights be, they are to one another as their 
base& [vi. i] 

Hence ab:bf= recL tu : rect bt 

= C:D 

= rect, ag ; rect bk. 

Therefore [vi. i, converse] the rectangles ag, bk have the same height, so 
that i is on the str^ght line hg. 

Hence A -^ B\B- recL ae ; rect. bt 

= af:bf 

= rect ak : rect bJi 
= C+D:D. 

SMondiy, let the magnitudes A, B'^ straight lines and the magnitudes 
C, i> areas. 

Let ab, if \x equal to the straight lines A, B, and to ab, bf apply the 
rectangles ag, bk equal to C, D respectively. 

Then, as before, the rectangles ag, bk have the same height. 

Now A + B:B^a/.bf 

= rect. aJk : rect. bk 

= C + D:D. i ,_ , ,. 

Thirdly, let all the magnitudes be straight lines. 

Apply to the straight lines C, D rectangles F, Q having the same height. 



<t < 








b 


J 


k t 


1 k 



^4 





BOOK V 


[v. 18, 19 


Then 


P:Q=C:D. 


[v.. l] 


Hence, by 


the second case, 




Also 


P+Q:Q=C+D.D. 




Therefore 


A + B:B = C+I):£>. 




11 


Proposition 19 


-., ' 



If f as a whoU is to a whole, so is a part subtracted to a 
part subtracted, the remainder will also be to the remainder 
as whole to whole. 

For, as the whole AB is to the whole CD, so let the 
part AE subtracted be to the part CF 
subtracted ; 

I say that the remainder EB will also be ■ ? 

to the remainder FD as the whole AB to c ^ d 
the whole CD. 

J 

For since, as AB is to CD, so is AE 
to CF, 
alternately also, as BA is to AE, so is DC to CF. [v. 16] 

And, since the magnitudes are proportional componendo, 
they will also be proportional separando, [v, 17] 

that is, as BE is to BA^ so is DF to CF^ 
and, alternately, 

as BE is to DF, so is EA to FC. [v, ifi] 

But, as AB is to CF, so by hypothesis is the whole AB 
to the whole CD. 

Therefore also the remainder EB will be to the remainder 
FD as the whole AB is to the whole CD. [v. 1 1] 

Therefore etc. 

[PoRisM. From this it is manifest that, if magnitudes be 
proportional componendo, they will also be proportional 
eonvertendo^ 

p. E. D. 

Algebraically, \l a:b = c\d (where e<.a and d < fi), then 
{a-<;):{i-d) = a:i. 

The " Porism " at the end of this projwsition is led up to by a few lines 
which Heiberg brackets because it is not Euclid's habit to explain a 
Porism, and indeed a Porism, from its very iwture, should not need any 



V. 19, so] PROPOSITIONS i8— ao it^ 

explanation, being a sort of by-product appearing without effort or trouble, 
air/nYfioTtvruf (Proclus, jpt 303, 6). But Heiberg thinks that Simson does 
wrong in finding fault with the argument leading to the "Porism," and that 
it does contain the true demonstration of conversion of a ratio. In this it 
appears to me that Heiberg is clearly mistaken, the supposed proof ott the 
basis of Prop. 19 being no more correct than the similar attempt to prove the 
inversion of a ratio from Prop. 4. Thu words are : " And since it was 
proved that, as A£ is to CB, so is EB to FJD, 

alternately also, as AB is to BE, so is CD to FD : 

therefore magnitudes when compounded are proportional. ,., 

But it was proved that, as £A is to AE, so is DC to CE and this is 
foniferiendo." 

It will be seen that this amounts to proving /ram the hypothesis a:b = €\d 
that the following transformations are simultaneously true, viz. : 

&\a~e-=b:h~d, 

and a'.c-b:d. 

The former is not proved from the latter as it ought to be if it were intended 
to prove conversion. 

The inevitable conclusion is that both the "Porism" and the argument 
leading up to it are interpolations, though no doubt made, as Heiberg says, 
before Theon's time. 

The conversion of ratios does not depend upon v. rg at all but, as Simson 
shows in his Proposition E {containing a proof already given by Clavius), on 
Props. 17 and 18. Prop. E is as follows. 

If four magnitudes U proportionals^ they are also proportionals by conversion, 
that is, tht first is to its excess above the second as the third is to 
its excess above tht fourth. ' 

Let .^J be to BE as CD to DF: 
then BA hloAE?ts DC to CE ^ ° 

Because AB is to BE as CD to DF, F 

by division [s^arando], 

AE'is to EB as CEto FD, [v. ij] 

and, by inversion, 

BE is to EA as DE to EC. 

[Simson's Prop, B directly obtained from v, Def. 5] 
Wherefore, by composition [com^nendo], 

BA is to AE as DC to CE [v. 18] 



Proposition 20. 

If there be three magnitudes, and others equal to them in 
multitttde, which taken two and two are in the same ratio, and 
if ^x aequali the first be greater than the third, the fourth will 
also be greater than the sixth; if equal, equal; and, if less, less. 



ff6 BOOK V [v. lo 

Let there be three magnitudes A, B, C, and others 
D, £, F equal to them in multitude, which taken two and 
two are in the same ratio, so that, 

■ as /^ is to J3t so is D to E, •■ 
and as B is to C, so is ^ to i^; 

and let y4 be greater than C ex aequali ; 
I say that D will also be greater than F\ \i A is equal to C, 
equal ; and, if less, less. 



A o- 

B E- 

c— F- 



For, since A is greater than C, 
and B is some other magnitude, 

and the greater has to the same a greater ratio than the less 
has, [v. 8] 

therefore A has to ^ a greater ratio than C has to B. 

But, as ^ is to B, so is D to E, 
and, as C is to B, inversely, so is /^ to ^ ; 
therefore Z? has also to ^ a greater ratio than 7^ has to ^, [v. 13] 

But, of magnitudes which have a ratio to the same, that 
which has a greater ratio is greater ; [v. 10] 

therefore D is greater than F. 

Similarly we can prove that, if j4 be equal to C, D will 
also be equal to F ; and if less, less. 

Therefore etc. 

Q. E. D. 

Though, as already remarked, Euclid ha£ not yet given us any definition 
of cempoanded ratios. Props, 20 — 23 contain an important part of the theory 
of such ratios. The term "compounded ratio" is not used, but the propositions 
connect themselves with the definitions of ex atguali in its two forms, the 
ordinary form defined in Def. 1 7 and that called ptrturbid proportion in 
Def. 18. The compounded ratios dealt with in these propositions are those 
compounded of successive ratios in which the consequent of one is the 
antecedent of the next, or the antecedent of one is the consequent of 
the next. 

Prop. Z2 states the fundamental proposition about the ratio tx aequali in 
its ordinary form, to the effect that, 

if a is to ^ as ^ is to f, 

and ^ is to If as « is to/, 

then a is to ^ as ^ is to/ 



V. ao] PROPOSITION 20 t JJ 

with the extension to any number of such ratios ; Prop, 23 gives the 
corresponding theorem for the case al perturbed proportion, namely thati 

if a is to ^ as « is to ^ .,^i 

and j is to ^ as </ is to e, 

then a IS to f as rf is to/ 

Each depends on a preliminary proposition. Prop, a a on Prop. 20 and 
Prop. 33 on Prop, a i. The course of the proof will be made most clear by 
using the algebraic notation. 

The preliminary Prop, 20 asserts that, 
if a ■.b = d:e, 

and t:c-t:/, 

then, according as «> = <<■, ds- = <./. 
For, according as a is greater than, equal to, or less than c, 
the ratio o ; * is greater than, equal to, or less than the ratio ( : b, [v, 8 or v. 7] 
or (since d:e = a:b, 

and £:b=/:e) 

the ratio d:e\s greater than, equal to, or less than the ratio/; t, 

[by aid of V. 13 and v. 11] 
and therefore d is greater than, equal to, or less than/ [v. 10 or v. 9] 
It is next proved in Prop. 22 that, by v. 4, the given proportions can be 
transformed into 

ma : nb = md : tie, 
and nb : pc = ne : pf, 

whence, by v. 20, 

according as ww is greater than, equal to, or less than, pc, 
md is greater than, equal to, or less than^ 
80 that, by Def. 5, • ■ ■ - . ' 

a:e = d:/. 

Prop, 23 depends on Prop. 21 in the same way as Prop. 22 on Prop, ao, 
but the transformation of the ratios in Prop. 23 is to the following : 
(i) ma : mb = ne : ttf 

(by a double application of v. 1 5 and by v. 11), 
(a) mb '.nc ~md\nt 

(by V. 4, or equivalent steps), 
and Prop, a I is then used. ' '" . r . ..^ 

Simson makes the proof of Prop, 20 slightly more explicit, but the main 
difference from the text is in the addition of the two other cases which Euclid 
dismisses with " Similarly we can prove." These cases are ; 

"Secondly, let A be equal to C; then shall D be equal to F. 
Because A and C are equal to one another, 

j4 is to .5 as C is to jB. [v, 7] 

But ■ ' ' A \% ifi B a& D '^ vo E, ■'.'., 

and C is to .f as .F is to E, 

wherefore i? is to £■ as 7^ to E ; . 1 -i [v. 11] 

and therefore Z> is equal to .^ ' 1 • .• [v. g] 



•..1.1 



ifg BOOK V [v. ao, « 

Next, let A be less than C; then shall JJ be less than J^. 
For C is greater than A, ■ ' - 

and, as w^ shown in the lirst case, 

Cis to Jas i^to^, 
and, in like manner, 

J is to vSt as ^ to i? ; 

therefore F is greater than D, by the first case ; and therefore D is less 
than JK" 

Proposition 21. 

// there be three magnitudes, and others equal to them in 
multitude, which taken two and two together are in the same 
ratio, and the proportion of them be perturbed, then, if ex 
aequali the first magnitude is greater than ths third, the 
fourth will also be greater than the sixth ; if equal, equal; 
and if less, less. 

Let there be three magnitudes A, B, C, and others D, E, F 
equal to them in multitude, which taken two and two are In 
the same ratio, and let the proportion of them be perturbed, 
so that, 

as A is to i?, so is ^ to F, , 

and, as ^ is to C, so is /? to E, 

and let A be greater than C ex aequali ; 

I say that D will also be greater than F\ if A is equal to 

C, equal ; and if less, less. 



A D- 

B^ — E- 

o F- 



For, since A is greater than C, 
and B is some other magnitude, 
therefore A has to ^ a greater ratio than C has to B. [v. 8] 

But, as A is to B, so is E to F, 
and, as C is to B, inversely, so is A to D. 
Therefore also E has to ^a greater ratio than E has to JD. 

[v-3] 

But that to which the same has a greater ratio is less ; 

[v. 10] 

therefore F is less than D ; 

therefore Z? is greater than F, 



V. 31, ijj PROPOSITIONS ao— 22 ^^ 

Similarly we can prove that, f 

if ^ be equal to C, D will also be equal to F\ 
and if less, less. 

Therefore etc. q. e. d. 

Algebraically, if a:b = e:f, 

and b:c=d\€, 

then, according asa> = <f, ;/> = <f. 
Simson's alterations correspond to those which he makes in Prop, a a. After 
the first case he proceeds thus. 

"Secondly, let A be equal to C; then shall D be equal to F. 
Because A and Care equal, 

H,i is to ^ as C is to ^. [v. 7] 

But /4 is to ^ as £ is to .^ 

and C is to .# as £ is to ^ : 

wherefore E is to /"as E to D, [v. 11] 

and therefore D is equal to F. [v. 9] 

Next, let A be less than C\ then shall D be less than F. 
For C is greater than A^ 
and, as was shown,* 

C is to .5 as .E to 2?, 
and, in like manner, 

.ff isto^as F\.aE\ 

therefore i^is greater than D, by the first case, 
and therefore D is less than F'^ 
The proof may be shown thus. 

According as (J > = < f, a;h> = <(;h. ' 
But a:b = e:fy and, by inversion, c:b = t:d. 

Therefore, according as «> = <<:, e:/> = <e:d, 
and therefore d> = </. 

Proposition 22. 
1/ there be any number 0/ magniiudes whatever, arui others 
equal to them in multitude, which taken two and two together 
are in the same ratio, they will also be in the same ratio ex 
aequali. 

Let there be any number of magnitudes A, B, C, and 
others Z>, £, F equal to them in multitude, which taken two 
and two tc^ether are in the same ratio, so that, 

as j4 is to .5, so is Z* to E^ 
and, as .5 is to C, so is £" to ^; 

I say that they will also be in the same ratio ex aequali, 
< that is, as .^ is to C, so is D to F> . 



xStt it— BOOK V [v. 21 

For of A, D let equimultiples G, H be taken, 
and of B, E other, chance, equimultiples A', L ; 
and, further, of C, F other, chance, equimultiples J/, N. 



A B c- 

D E- — F- 

— I K 1 ► 



Then, since, as A is to B, so is Z? to £", 
and of A, D equimultiples G, H have been taken, 
and of B, E other, chance, equimultiples K, L, 

therefore, as 6^ is to K, so is H to L. [v. 4] 

For the same reason also, 

as A' is to M, so is L to N. 

Since, then, there are three magnitudes G, K, M, and 
others H, L, N equal to them in multitude, which taken two 
and two together are in the same ratiQ, 

therefore, ex aeguali, if G is in excess of M,H\s also in excess 
oiN; 

if equal, equal; and if less, less. ■ - .i [v. ao] 

And G, H are equimultiples o( A, D, 

and M, N other, chance, equimultiples of C, F. 

Therefore, as ^ is to C, so is D to F. [v. Def. 5] 

Therefore etc, 

Q. E. D. 

EucUd enunciates this proposition as true of any number of magnilvdti 
whatetier forming two sets connected in the manner described, but bis proof is 
confined to the case where each set consists of three magnitudes only. The 
extension to any number of magnitudes is, however, easy, as shown by 
Simson. 

"Next let there be four magnitudes A,B,C, D, and other four E, F, G, Jf, 
which two and two have the same ratio, viz. : 
as /4 is to ^, so is -£ to ./% 



A B C D 
E F O H 



and as £ is to C, so is .^to G, 

and as C is to A so is C to .ff ; 
A shall he to D as £ to IT. 
Because A, S, C are three magnitudes, and E, .f, G other three, which 
taken two and two have the same ratio, 
by the foregoing case, 

v^ is to C as £ to 6^. 



y. 31, *3) PROPOSITIONS ai, aj 

But C is to Z) as C is to If; 
wherefore again, by the fiist case, 

^ is to Z> as £ to /^ 
And so on, whatever be the number of magnitudes." 



Proposition 23. 

// there be three magnitudes, and others equal to them in 
mtdtilude, which taken two and two together are in the same 
ratio, and the proportion of them be perturbed, they will also 
be in the same ratio ex aequali. 

Let there be three magnitudes A, B, C, and others equal 
to them in multitude, which, taken two and two together, are 
in the same proportion, namely D, E, F\ and let the propor- 
tion of them be perturbed, so that, 

as >4 is to B, so is £" to F, 
and, as j9 is to C, so is Z? to ^ ; ' 

1 say that, as ^ is to C, so is Z? to .F. 

A B — c 

D E^ F- — - 

O 1 1 H 1 1 L ■ 

K 1 1 M 1 N' 1 



Of W, B, D let equimultiples G, H, Kh^ taken, 
and of C, E, Z^ other, chance, equimultiples L, M, N. 

Then, since G, //^are equimultiples oi A, B, 
and parts have the same ratio as the same multiples of 
them, [v. is] 

therefore, as A is to .5, so is G^ to H. 
For the same reason also, 

as .£■ is to /% so is i?/ to A^. 
And, as .^ is to B, so is E to E\ 

therefore also, as G is to H, so is M to N. [v. 11] 

Next, since, as ^ is to C, so is D to E, 
alternately, also, as B is to D, so is C to E. [v. i«] 

And, since H, K are equimultiples of B, D, 
and parts have the same ratio as their equimultiples, 

therefore, as .f is to Z?, so is Z^ to K. [v. i s] 



tti tA • BOOK V [v. 23 

But, as ^ is to Z7, so is C to £' ; 
therefore also, as // is to A', so is C to B, [v. n] 

Again, since L, M are equimultiples of C, E, 

therefore, as C is to E, so is L to jIT. [v. 15] 

But, as C is to E, sovsHtoK; 

therefore also, as H is to K, so is L to M, (v. 1 1] 

and, alternately, as //^ is to Z, so is A' to M. [v. 16] 

But it was also proved that, 

^s G \s to H, so is M to A''. 

Since, then, there are three magnitudes G, H, L, and 
others equal to them in multitude K, M, N, which taken two 
and two together are in the same ratio, 
and the proportion of them is perturbed, 
therefore, ex aequalif if G is in excess of L, K is also in excess 
of A''; , ., , ^., 

if equal, equal; and if less, less. [v. n] 

And G, K are equimultiples of A, D, 
and L, N oi C, F. 

Therefore, as A is to C, so is Z? to /^ 

Therefore etc. 

Q. E. D. 

There b an important difference between the version given by Simson of 
one part of the proof of this proposition and that found in the Greek text of 
Heiberg. Peyrard's ms. has the version given by Heiberg, but Simson's 
version has the authority of other mss. The Basel editw prinaps gives both 
versions (Simson 's being the first). After it has been proved by mean? of 
V. 1 5 and V. 1 1 that, 

as G is to jff, so is jV to A^ 
or, with the notation used in the note on Prop, *o, 

ma \ mb = ne ; nf, 
it has to be proved further that, 

• as .ff is to Z, so is /T to M, 
or mb •.nc = md : ne, 

and it is clear that the latter result may be directly inferred from v. 4, The 
reading translated by Simson makes this inference : 

" And because, as £ is to C, so is Z? to £, 
and H, K sxt equimultiples of S, /?, 
and L, Mot C, E, 

therefore, as H is to Z, so is K to M" [v. 4] 

The version in Hei berg's text is not only much longer (it adopts the 

roundabout method of using each of three Propositions v. 11, 15, 16 twice 



V. 33. S4] PROPOSITIONS 33, 24 183 

over), but it is open to the objection that it uses v, 1 6 which is only applicable 
if the four magnitudes are of the same kind; whereas v. 33, the proposition 
now in question, is not subject to this restriction. 

Simson rightly observes that in the last step of the proof it should be 
stated that " G, K are any equimultiples whatever of A, D, a.nA L, N any 
whatever of C, F." 

He also gives the extension of the proposition to any number of magnitudes, 
enunciating it thus 1 

" If there be any number of magnitudes, and as many others, which, taken 
two and two, in a cross order, have the same ratio ; the first shall have to the 
last of the first magnitudes the same ratio which the first of the others has to 
the last " ; -,.••,,'. 

and adding to the proof as follows : \ . t < 1 

"Next, let there be four magnitudes A, B, C, D, and other four E, F, G, If, 
which, taken two and two in a cross order, have the same ratio, viz. : 
Am BasGto H, 



Bto Cas FioG, A B C 

-W6A CtoDasM toF; | E F Q H 

then A is to J} a& £ to M. 

Because A, B, C are three magnitudes, and F, G, H other three which, 
taken two and two in a cross order, have the same ratio, 

by the first case, .^ is to C as ^to H. Ii ^ <• 

But C is to .0 as £ is to F\ 

wherefore again, by the first case, . , 

A\s,\oDi&E\oH. 
And so on, whatever be the number of magnitudes." 



Proposition 24. 

If a first magnitude have to a second the same ratio as a 
third has to a fourth, and also a fifth have to the second the 
same ratio as a sixth to the fourth, the first and fifth added 
together milt have to the second the same ratio as the third and 
sixth have to the fourth. 

Let a first magnitude AB have to a second C the same 
ratio as a third DE has to a 

fourth F; f^ — : g q 

and let also a fifth BG have to o 

the second C the same ratio as d 1 H 

a sixth EJ/ has to the fourth f 

E; 

1 say that the first and fifth added together, AG, will have 
to the second C the same ratio as the third and sixth, I? J/, 
has to the fourth E. 



1*4 BOOK V [v. 34 

For since, as BG is to C, so is B// to F, 
inversely, as C is to BG, so is ^ to BH. 

Since, then, as AB is to C, so is DB to B, 
and, as C is to BG, so is F to B//, 
therefore, ex aequali, as AB is to ^G, so is DB to ^/^. [v. n] 

And, since the magnitudes are proportional separando, they 
will also be proportional componendo ; [v. 18] 

therefore, as AG is to GB, so is Z?/^ to HB. 

But also, as BG is to C, so is BH to A ; 
therefore, ex aegtta/i, as ^4 6" is to C, so is DH to F. [v. aa] 

Therefore etc. Q. e. d. 

Algebraically, if a -.c = d:f, 

and b\(~e:f, 

then ■• {a-\-b):c={d-¥i):J. 

This profwsition is of the same character as those which precede the 
propositions relating to compounded ratios ■ but it could not be placed earlier 
than it ts because v. 22 is used in the proof of it. 

Inverting the second proportion to 

c\b-f:t, 
it follows, by v, 23, that a;6 = d:t, 

whence, by v. 18, {a + b):i ~{d-i-e) : e, 

and from this and the second of the two given proportions we obtain, by a 
fresh application of v. 22, 

{a-^l,):c=(d*e):/. 

The first use of v. as is important as showing that the opposite process to 
compounding ratios, or what we should now call division of one ratio by 
another, does not require any new and separate propositions. 

Aristotle tacitly uses v. 24 in combination with v. 1 1 and v, 16, Meleorologica 
'"- S> 37^a 22 — 26. 

Simson adds two corollaries, one of which (Cor. 3) notes the extension to 
any number of magnitudes. 

" The proposition holds true of two ranks of magnitudes whatever be their 
number, of which each of the first rank has to the second magnitude the same 
ratio that the corresponding one of the second rank has to a fourth magnitude ; 
as is manifest" 

Simson's Cor. i states the corresponding proposition to the above with 
separando taking the place of compomnds, viz., that corresponding to the 
algebraical form 

{a-b);c^{fi-e):f. 

"Cor. I. If the same hypothesis be made as in the proposition, the 
excess of the flrst and fifth shall be to the second as the excess of the third 
and sixth to the fourth. The demonstration of this is the same with that of 
the proposition if division be used instead of composition." That is, we use 
V. 17 instead of v. 18, and conclude that 

{a~b):b = {d-t):t. 



V. as] PROPOSITIONS »4, »s ^ 

Proposition 25. 

If four magnitudes be proportional, the greatest and the 
least are greater than the remainittg two. 

Let the four magnitudes A£, CD, E, F be proportional 

so that, as AB is to CD, so is E to 

F, and let AB be the greatest of them 

and F the least ; ^ Q b 

I say that AB, F are greater than c 

CD, E. H P 

c 1 — 

For let AG be made equal to E, , 

and CH equal to F, 

Since, as ^^ is to CD, so is ^ 
to F, 

and E is equal to AG, and /^ to CM, 

therefore, as AB is to CD, soh AG to C/T. 

And since, as the whole AB is to the whole CD, so is 
the part AG subtracted to the part C// subtracted, 

the remainder GB will also be to the remainder HD as 
the whole AB is to the whole CD. [v. 19] 

But AB is greater than CD ; 
therefore GB is also greater than HD. 

And, since AG\& equal to E, and CH to F, 
therefore AG, F a.re equal to CH, E. 

And if, GB, HD being unequal, and GB greater, AG, F 
be added to GB and C^, ^ be added to HD, 

it follows that AB, F are greater than CD, E. 

Therefore etc. 

Q. E. D. 

Algebraically, if a-.b^cid, 

and a is the greatest of the four magnitudes and d the least, 

a-¥ d> 6 + c. 

Simaon is right in inserting a word in the setting-out, "let AB be the 
greatest of Ihem and <censequenily> J' the least." This follows from the 
particular case, really included in Def. 5, which Sinison makes the subject of 
his proposition A, the case namely where the equimultiples taken are ante the 
several magnitudes. 

The proof is as follows. 

Since a:b = £:d, 

a — {\h — d=a:b, [v. 19] 



iM BOOK V [v. 25 

But «>*; therefore (tf-^)>(*-i/). • [v. 16 and 14] 

Add to each {c+d); 
therefore (o + rf) > (i + c). 

There is an important particular case of this proposition, which is, 
however, not mentioned here, vh. the case where * = c. The result shows, in 
this case, that tAe arithmttic PKan between two magnitudes is greater than 
ffielr geometric mean. The truth of this is proved for straight lines in vi.- 27 
by "geometrical algebra," and the theorem forms the Siopur/ioj for equations 
of the second degree. 

Simson adds at the end of Book y, four propositions, F, G, H, K, which, 
however, do not seem to be of sufficient practical use to justify their inclusion 
here. But he adds at the end of his notes to the Book the following 
paragraph which deserves quotation word for word. 

"The 5th book being thus corrected, I most le.idily agree to what the 
learned Dr Barrow says, 'that there is nothing in ihe whole boily of the 
elements of a more subtile invention, nothing more solidly established, and 
more accurately handled than the doctrine of proportionals.' And there is 
some ground to hope that geometers will think that this could not have been 
said with as good reason, since Theon's time till the present." 

Simson's claim herein will readily be admitted by all readers who are 
competent to form a judgment upon his criticisms and elucidations of Book V. 



BOOK VI. 



INTRODUCTORY NOTE. 

The theory of proportions has been established in Book v. in a perfectly 
general form apphcable to all kinds of magnitudes (although the representation 
of magnitudes by straight lines gives it a j^eo metrical appearance) ; it is now 
necessary to apply the theory to the particular case oi geometrical investigation. 
The only thing still required in order that this may be done is a proof of the 
existence of such a magnitude as bears to any given finite magnitude any 
given finite ratio ; and this proof is supplied, so far as regards the subject 
matter of geometry, by vi. n which shows how to construct a fourth pro- 
portional to three given straight lines, 

A few remarks on the enormous usefulness of the theory of proportions 
to geometry will not be out of place. We have already in Books i. and ii. 
made acquaintance with one important part of what has been well called 
geometrical algebra, the method, namely, of application of areas. We have 
seen that this method, working by the representation of products of two 
quantities as rectangles, enables us to solve some particular quadratic equations. 
But the limitations of such a method are obvious. So long as general 
quantities are represented by straight lines only, we cannot, if our geometry 
is plane, deal with products of more than two such quantities ; and, even 
by the use of three dimensions, we cannot work with products of more 
than three quantities, since no geometrical meaning could be attached to 
such a product. This limitation disappears so soon as we can represent any 
general quantity, corresponding to what we denote by a letter in algebra, by 
a ratio; and this we can do because, on the general theory of proportion 
established in Book v., a ratio may be a ratio of two incommensurable 
quantities as well as of com mensu rabies. Ratios can be compoundeti ad 
infinitum, and the division of one ratio by another is equally easy, since it is 
the same thing as compounding the first ratio with the inverse of the second. 
Thus e,g. it is seen at once that the coefficients in a quadratic of the most 
general form can be represented by ratios between straight lines, and the 
solution by means of Books i. and n, of problems corresponding to quadratic 
equations with particular coefficients can now be extended to cover any 
quadratic with real roots. As indicated, we can perform, by composition of 
ratios, the operation corresponding to multiplying algebraical quantities, and 
this to any extent. We can divide quantities by compounding a ratio with 
the inverse of the ratio representing the divisor. For the addition and 
subtraction of quantities we have only to use the geometrical equivalent of 
bringing to a common denominator, which is effected by means of the fourth 
proportional. .. ^ . , _ 



i8» BOOK VI [vi. DErr. 



DEFINITIONS. 

I. Similar rectilineal figures are such as have their 
angles severally equal and the sides about the equal angles 
proportional. 

[2. Reciprocally related figures. See noie.'\ ■'■•'' 

3. A straight line is said to have been cut in extreme 
and mean ratio when, as the whole line is to the greater 
segment, so is the greater to the less. 

4. The height of any figure is the perpendicular drawn 
from the vertex to the base. 



Definition i. 

QfiGija. (Tj^^^ara tv&vypofifid fiTTtv, wra rav re ymvia^ l^tK i\Vr Kara fjttav ital 
T&f TTtpl Tat tirat '^ott'taf frX^pa^ (ikoAd^OK 

T'lis definition is quoted by Aristotle, Ana/, post. 11. 17, 99 a 13, where 
he says that simUantjf (to o;u.Diof) in the case of figures "consists, let us say 
(htuk), in their having their sides proportional and their angles equal." The 
use of the word laat may suggest that, in Aristotle's time, this definition had 
not quite established itself in the text-books (Heibeig, MathemaiUches zu 
Arisiolties, p. g). 

It was pointed out in Van Swinden's Eknunts of Geometry (Jacobi's 
edition, 1834, pp. 1 14 — 5) that Ei^uclid omits to stale an essential part of the 
definition, namely that "the corresponding sides must be opposite to equal 
angles," which is necessary in order that the corresponding sides may follow 
in the same order in both figures. 

At the same time the definition states more than is absolutely necessary, 
for it is true to say that iwo polygons are similar when, if the iides and angles 
are taken in the same order, the angles are equal and the sides about the equal 
angles are proportional, omitting 

(i) three consecutive angles, 

or (2) two consecutive angles and the side common to them, 

or (3) two consecutive sides and the angle included by them, 

and making no assumption with regard to the omitted sides and angles. 

Austin objected to this definition on the ground that it is not obvious that 
the properties (i) of having their angles respectively equal and (2) of having 
the sides about the equal angles proportional can coexist in two figures ; but, 
a definition not being concerned to prove the existence of the thing defined, 
the objection falis to the ground. We are property left to satisfy ourselves as 
to the existence of similar figures in the course of the exposition in Book vi., 
where we learn how to construct on any given straight line a rectilineal figure 
similar to a given one (vi. i8j. 



VI. DEFF. a— s] DEFINITIONS 189 

Definition 2. 

The Greek text gives here a. definition of riciprocally related fibres 
(dtTTnrtTTOV^d™ ay^^T<i\. "[Two] figures are redprocally relaUd when there 
are in each of the two figures antecedent and consequent ratios" ('An-HrnrokSora 
Si o-j()j/iaTa i<mv, oray tv iKorifxf Tiui' (j}(T)iiaTiati ijyiivfio'oi t€ nai Iro/uTOt Aoyoi 
wiTtv). No intelligible meaning can be attached to "antecedent and con- 
sequent ratios " here ; the sense would require rather " an antecedent and a 
consequent of (two equal) ratios in each figure." Hence Candalla and 
Peyrard read Xoywc Spot ("terms of ratios") instead of Ao'yiii. Camerer reads 
Xayar without upoi. But the objection to the definition lies deeper. It is 
never used; when we come, in vi. 14, 15, xi. 34 etc. to jjaiallelograms, 
triangles etc. having the property indicated, they are not called " reciprocal " 
parallelograms etc., but parallelograms etc. "/A^ sides ofwhUh are reciprocally 
proportional," w/ ot-r(irtiroi'Sa<ric at irAtupoi, Hence Sim son appears to be 
right in condemning the definition; it may have been interpolated from Heron, 
who has it. 

Simson proposes in his note to substitute the following definition. "Two 
magnitudes are said to be reciprocally proportional to two others when one 
of the first is to one of the other magnitudes as the remaining one of the last 
two is to the remaining one of the first." This definition requires that the 
magnitudes shall be all of the same kind. 



Definition 3. 

'Afipav nat, fiiirov Aoyoi' tiStia Tir/iigcrAu Xiytmt, Srav ^ wt 7 Skii -rp^ rd 
fut^otf T/jiTJ/juij oi/ru? T^ fArti^ov irpof ra IXaTTCv* 



Definition 4. 

'Ayofimj^ 

The definition of " height " is not found in Campanus and is perhaps 
rightly suspected, since it does not apply in terms to parallelograms, parallele- 
pipeds, cylinders and prisms, though it is used in the Elements with reference 
to these latter figures. Aristotle does not appear to know altitude (v^ot) in 
the mathematical sense; he uses naStTiK of triangles (Meiearelo^ea tn. 3, 
373 a 11). The term is however readily understood, and scarcely requires 
definition. 

[Definition j. 

Aoyoc Ik koywv (TvyKturBai XcycTat, crap at twv koytav injAuroTi/rc; j0* lavrac 
woXXankatTiauSturat vtumtrl riva. 

"A ratio is said to be compounded of ratios when the sizes (jnjAiKortjrt?) of 
the ratios multiplied together make some (? ratio, or size),"] 

As already remarked (pp. 116, 132), it is beyond doubt that this definition 
of ratio is interpolated. It has little MS. authority. The best MS. (P) only has 
it in the margin; it is omitted altogether in Campanus' translation from the 



ijli* BOOK VI [vi. OEF. s 

Arabic ; and the other mss. which contain it do not agree in the position 
which they give to it. There is no reference to the definition in the place 
where compound ratio is mentioned for the first time (vi. t^), nor anywhere 
else in Euclid; neither is it ever referred to by the other great geometers, 
Archimedes, Apollonius and the rest. It appears to be only twice mentioned 
at all, {:) in the passage of Eutocius referred to above (p. ii6) and (») by 
Theon in his commentary on Ptolemy's ffun-o^u. Moreover the content of 
the definition is in itself suspicious. It speaks of the " sizes of ratios being 
multiplied together (literally, into themselves)," an operation unknown to 
geometry. There is no wonder that Eutocius, and apparently Theon also, in 
their efforts to explain it, had to give the word jrrfAiiHjnjt a meaning which has 
no application except in the case of such ratios as can be expressed by 
numbers (Eutocius e.g. making it the "number by which the ratio is called"). 
Nor is it surprising that Wallis should have found it necessary to substitute 
for the " quantitas " of Commandinus a different translation, " quantuplicity," 
which he said was represented by the "expeneni af the ratio" ( ratio nis ex- 
ponens), what Peletarius had described as "denominatio ipsae pro portion! s" 
and Clavius as "denominator." The fact is that the definition is ungeometrical 
and useless, as was already seen by Savile, in whose view it was one of the 
two blemishes in the body of geometry (the other being of course Postulate 5). 
It is right to add that Hultsch (art. "Eukleides" in Pauly-Wissowa's Real- 
EtuytlopddU dtr danischen Allertumswissenschaft) thought the definition 
genuine. His grounds are (i) that it stood in the iroAaui ln&ooi^ repre- 
sented by P (though P has it in the margin only) and (a) that some ex- 
planation on the subject must have been given by way of preparation for 
VI. 25, while there is nothing in the definition which is incomisteni with the 
mode of statement of vi. 23. If the definition is after all genuine, I should 
be inclined to regard it as a mere survival from earlier textbooks, like the first 
of the two alternative definitions of a solid angle (xt, Def 11); for its form 
seems to suit the old theory of proportion, applicable to commensurable 
magnitudes only, better than the generalised theory of Eudoxus, 






v. kI|.'-' . ..i \-!\' 



.... ;*-,, r,. jV 1,'r.i' -■ ' '" 



BOOK VI. PROPOSITIONS. 



Propositiok 



'•> J MiS' 



Triangles and parallelograms which are under the same 
height are to one another as their bases. 

Let ABC, A CD be triangles and EC, C/^ parallelograms 
under the same height ; 

j I say that, as the base BC is to the base CZ7, so is the 
triangle ABC to the triangle A CD, and the parallelogram 
£C to the parallelogram CF. 




For let BD be produced in both directions to the points 
ff, L and let [any number of straight lines] BG, GH be 
lo made equal to the base BC, and any number of straight lines 
DK, KL equal to the base CD ; 

let AG, AH, AK, AL be joined. 

Then, since CB, BG, GH are equal to one another, 
the triangles ABC, AGB, AHG are also equal to one 
IS another. [i. 38] 

Therefore, whatever multiple the base HC is of the base 

BC, that multiple also is the triangle AHC of the triangle 

ABC. 

For the same reason, 
10 whatever multiple the base ZC is of the base CD, that 
multiple also is the triangle ALC of the triangle ACD ; 
and, if the base HC is equal to the base CL, the triangle 
AHC is also equal to the triangle ACL, [i. 38] 



r9» BOOK VI [n. i 

if the base //C is in excess of the base CZ., the triangle AHC 
as is also in excess of the triangle A CL, 
and, if less, less. 

Thus, there being four magnitudes, two bases BC, CD 
and two triangles ABC, ACD, 

equimultiples have been taken of the base BC and the 
30 triangle ABC, namely the base HC and the triangle AHC, 
and of the base CD and the triangle ^Z?C other, chance, equi- 
multiples, namely the base LC and the triangle ALC \ 

and it has been proved that, 
if the base HC is in excess of the base CL, the triangle AHC 
3S is also in excess of the triangle ALC ; 
if equal, equal ; and, if less, less. 

Therefore, as the base BC is to the base CD, so is the 

triangle ABC to the triangle ACD. [v. Def. 5] 

Next, since the parallelogram EC is double of the triangle 

AoABC, [i. 4>] 

and the parallelogram FC is double of the triangle ACD, 

while parts have the same ratio as the same multiples of 

them, [v. 15] 

therefore, as the triangle ABC is to the triangle ACD, so is 

45 the parallelogram £C to the parallelogram FC. 

Since, then, it was proved that, as the base BC is to CD, 
so is the triangle ABC to the triangle A CD, 
and, as the triangle ABC is to the triangle ACD, so is the 
parallelogram EC to the parallelogram CF, 
50 therefore also, as the base BC is to the base CD, so is the 
parallelogram EC to the parallelogram FC. [v. n] 

Therefore etc. 

Q, E. D. 

4. Under the same height. The Greek text has "under Ihe iame height AC," with 
a figure in which the side ^C commun to the two triangles is perpendicular to the base and 
is therefore iudf the "height." But, even if tlie two triangles are placed contiguously so as 
to have a commbn side AC, it is quite gratuiicnis to require it to be perpendicular to the base. 
Theon, on this occasion making an improvement, altered to " which are [Ato) under the 
same height, (namely! 'he perpendicular drawn from A to BD," I iiave vetitured lo alter so 
far as lo omit "AC" and to draw the figure in the usual way. 

14. ABC.AGBiAHG. Euclid,indiaferenttoeitactorder,writes" AffG, AGB,ABC." 
46. Since then it was proved that, as the base BC i( to CD, *o It the triangle 
ABC lo the triangle ACD. Here again words have to be supplied in translatin|> the 
eitremely terse Greek irtl a^ ^Htlx^fj "• 1'^' ^ fii'" Br rpit T^e Ti, otrut ri ABr 
Tf/iyuror r^i ri ATA Tplywrwr, literSly " since was proned, as the base BC to CO, to the 
truaigle ABC lo Ihe triangle ACJi." Cf. note on v. 16, p. i6j. 



VI. i] PROPOSITION I 193 

The proof assumes — what is however an obvious deduction from 1. 38 — 
that, of triangles or parallelograms on unequal bases and between the same 
parallels, the greater is that which has the greater base. 

It is of course not necessary that the two given triangles should have a 
common side, as in the figure ; the proof is just as easy if they have not. 
The proposition being equally trtie of triangles and parallelograms of eqital 
heights, Simson states this fact in a corollary thus: 

" From this it is plain that triangles and parallelograms that have equal 
altitudes are to one another as their bases. 

Let the figures be so placed as to have their bases in the same straight 
line ; and, if we draw perpendiculars from the vertices of the triangles to the 
basw, the straight line which joins the vertices is parallel to that in which 
their bases are, because the perpendiculars are both equal and parallel to one 
another [i, 33]. Then, if the same construction be made as in the proposition, 
the demonstration will be the same." 

The object of placing the bases in one straight line is to get the triangles 
and parallelograms within (hi same parallels. Cf. Proclus' remark on i. 38 
(p. 405, 17) that having the same height is the same thing as being in the 
same parallels. 

Rectangles, or right-angled triangles, which have one of the sides about 
the right angle of the same length can be placed so that the equal sides 
coincide and the others are in a straight line. If then we call the common 
side the base, the rectangles or the right-angled triangles are to one another 
as their heights, by vi. i. Now, instead of each right-angled triangle or 
rectangle, we can take any other triangle or parallelogram respectively with an 
equal base and between the same parallels. Thus 

Triangles and paralklograms having eguai bases art to one another as their 
heights. 

Legendie and those authors of tnodem text-books who follow him in 
basing their treatment of proportion on the algebraical definition are obliged 
to divide their proofs of propositions like this into two parts, the first of 
which proves the particular theorem in the case where the magnitudes are 
commensurable, and the second extends it to the case where they are 
incommensurable. 

L^endre (Aliments dt G'eometrie, \\\. 3) uses for this extension a rigorous 
method by reductie ad absurdum similar to that 
used by Archimedes in his treatise On the 
equilibrium 0/ planes^ 1. 7. The following is 
Legendre's proof of the extension of vi, i to in- 
commensurable parallelograms and bases. 

The proposition having been proved for 
commensurable bases, let there be two rectangles 
AS CD, AEFD as in the figure, on bases AB, 
.^£which are incommensurable with one another. 

To prove that recL A BCD: recL AEFD =AB: AE. 

For, if not, let red. A3CD -rect. AEFD = AB : AO, (ij 

where AO h (for instance) greater than AE. 

Divide AS into equal parts each of which is less than EO, and mark off 
on AO lengths equal to one of the parts; then there will be at least one point 
of division between E and O. -r 

Let it be /, and draw /AT parallel to EF. * 




«94 



BOOK VI [vi. 1, » 



Then ihe rectangles A BCD, AIKD are in the ratio of the bases AB, A I, 
since the latter are commensurable. 
Therefore, inverting the proportion, 

rect. AIKD:^<ixx. ABCD'^AI.AB (i). 

From this and (i), « atquati, 

rect, AIKD : rect. AEFD = AI.AO. , _ 

But A0> A/; therefore rect. AEBD>Ttci. AIKD. 
But this is impossible, for the rectangle AEFD is less than the rectangle 
AIKD. 

Similarly an impossibility can be proved \i AO < AE. > 

Therefore lecL ABCD : rect. AEFD = AB : AE. 

Some modern American and German text-books adopt the less rigorous 
method of appealing to the theory of iimiis. 



Proposition 2. 

If a straight line be drawn parallel to one of the sides of a 
triangle, it will cut the sides of the triangle proportionally ; 
and, if Ike sides of the triangle be cut proportionally, the line 
joining the points of section will be parallel to the remaining 
side of the triangle. 

For let DE be drawn parallel to BC, one of the sides of 
the triangle ABC; 

I say that, as BD is to DA, so is CE to 
EA. 

For let BE, CD be joined. 

Therefore the triangle BD E is equal to 
the triangle CDE ; 

for they are on the same base DE and in 
the same parallels DE, BC. [1. 38] 

And the triangle ADE is another area. 

But equals have the same ratio to the same ; [v. 7] 

therefore, as the triangle BDE is to the triangle ADE^ so 
is the triangle CDE to the triangle ADE, 

But, as the triangle BDE is to ADE, so is BD to DA ; 

for, being under the same height, the perpendicular drawn 
from E to AB, they are to one another as their bases, [vi. i] 

For the same reason also, 
as the triangle CDE is to ADE, so is CE to EA. '- 

Therefore also, as BD is to DA, so is CE to EA. [v. n] 




VI. a, 3] PROPOSITIONS 1—3 195 

Again, let the Sides AB, AC oi the triangle ABC be cut 

proportionally, so that, as BD is to DA, so is C£ to EA ; 

and let DE be joined. 

I say that DB is parallel to BC - f •" • 

For, with the same construction, i ■ •"> -^ • 

since, as BD is to DA, so is C£ to £A, 

but, as BD is to DA, so is the triangle BDE to the triangle 

ADE, 

and, as CE is to EA, so is the triangle CDE to the triangle 
ADE, [v.. ,] 

therefore also, 

as the triangle BDE is to the triangle ADE, so is the 
triangle CDE to the triangle ADE. [v. n] 

Therefore each of the triangles BDE, CDE has the same 
ratio to ADE. 

Therefore the triangle BDE is equal to the triangle CDE\ 

[V.9] 
and they are on the same base DE. 

But equal triangles which are on the same base are also 
in the same parallels. [i. 39] 

Therefore DE is parallel to BC. 
Therefore etc. 

Q, E. D. 

Euclid evidently did not think it worth while to distinguish in the 
enunciation, or in the figure, the cases in which the parallel to the base cuts 
the othei two sides produced (a) beyond the point in which they intersect, 
(i) m the other direction. Simson gives the three figures and inserts words 
in the enunciation, reading "it shall cut the other sides, or those lidts produced, 
proportionally" and "if the sides, or the sidts produced, be cut proportionally." 

Todhunter observes that the second part of the enunciation ought to 
make it clear which segments in the proportion correspond to which. Thus 
e.g., if AD were double of DB, and CE double of EA, the sides would be 
cut proportionally, but DE would not be parallel to BC. The omission 
could be supplied by saying "and if the sides of the triangle be cut 
proportionally io that tht segments adjacent to the third side «ri corresponding 
terms in the proportion." 

.....-.,■ i ' |« , . ..... .,. 

Proposition 3. . 

// an angle of a triangle be bisected and the straight line 
cutting the angle cut the base also, the segments of the base 
will have the same ratio as the remaining sides of the triangle; 
and, if ike segments of the base have the same ratio as the 




1^6 - BOOK VI [vi. 3 

remaining sides of the triangle, the straight line joined from 
the vertex to the point of section will bisect the angle of the 
triangle. 

Let ABC be a triar^le, and let the angle BA C be bisected 
by the straight line AD \ 

I say that, as BD is to CD, so 
is BA to AC. 

For letC^ be drawn through 
C parallel to DA, and let BA 
be carried through and meet it 

Then, since the straight line 
A C falls upon the parallels AD, 
EC, 

the angle ACE is equal to the angle CAD. [i, 39] 

But the angle CAD is by hypothesis equal to the angle 
BAD; 

therefore the angle BAD is also equal to the angle ACE. 

Again, since the straight line BAE falls upon the parallels 
AD, EC, 

the exterior angle BAD is equal to the interior angle 
A EC. [i. 39] 

But the angle ACE was also proved equal to the angle 
BAD; 

therefore the angle A CE is also equal to the angle A EC, 

so that the side AE is also equal to the side AC. [i. 6] 

And, since AD has been drawn parallel to EC, one of 
the sideij of the triangle BCE, 

therefore, proportionally, as BD is to DC, so is BA to AE. 

But AE is equal to AC; t^'- '] 

therefore, as BD is to DC, so is BA to AC. 

Again, let BA be to ^Cas BD to DC, and let AD be 
joined ; 

I say that the angle BAC has been bisected by the straight 
line AD. 

For, with the same construction, 
since, as BD is to DC, so is BA to AC, •'■ -^^ i' • ^<y-^ 



n.3] PROPOSITION 3 197 

and also, as BD is to DC, so is BA to AE\ for AD has 
been drawn parallel to EC, one of the sides of the triangle 
BCE : [VI. 2] 

therefore also, as BA is to A C, so is BA to AE. [v. i r] 

Therefore AC is equal to AE, [v. 9] 

so that the angle A EC is also equal to the angle ACE. [i. 5] 

But the angle A EC is equal to the exterior angle BAD, 

[I. jg] 
and the angle ACE is equal to the alternate angle CAD; ['d.] 

therefore the angle BAD is also equal to the angle CAD. 

Therefore the angle ^^C has been bisected by the straight 
line AD. 

Therefore etc. •' • 

Q, E. D. 

The demonstration assumes that C£ will meet BA produced in some 
point £. This is proved in the same way as it is proved in vi. 4 that BA, ED 
will meet if produced. The angles ABD, SDA in the figure of vc, 3 are 
together less than two right angles, and the angle BDA is equal to the angle 
BCE, since DA, CE are parallel. Therefore the angles ABC, BCE are 
together less than two right angles ; and BA, CE must meet, by 1. Post. 5. 

The corresponding proposition about the segments into which ^C is 
divided externally by the bisector of the external angU at A when that 
bisector meets BC produced (i.e. when the sides AB, AC ak not equal) is 
important. Simson gives it as a separate proposition, A, noting the fact that 
Pappus assumes the result without proof (Pappus, vii. p, 730, 24). 

The best plan ts however, as De Morgan says, to combine Props. 3 and A 
in one proposition, which may be enunciated thus : If an angle of a triangle 
be bisected internally or externally by a straight line which cuts the opposite side 
or the opposite side produced, the segments of that side will have the same ratio 
as the other sides of the triangle; and, if a side of a triangle be divided internally 
or externally so that its segments have the same ratio as the other sides of the 
triangle, the straight line drawn from the point of section to the angular point 
which is opposite to the first ?nentiomd side will bisect the interior or exterior angle 
at that angular point. 





Let ..4 C be the smaller of the two sides AB, AC, so that the bisector AD 
of the exterior angle at A may meet BC produced beyond C. Draw CE 
through C parallel to DA, meeting BA in E. 

Then, if EA C is the exterior angle bisected by AD in the case of external 
bisection, and if a point Eis taken on AB io the figure of vi. 3, the proof of 




t9». BOOK VI [vi. 3 

VL 3 can be used almost word for word for the other ease. We have only to 
spak of the angle "/^^^C" for the angle " BAC," and of the angle "FAD" 
for the angle " BAD " wherever they occur, to say "let SA^ or BA produced, 
meet CE in E," and to substitute " BA or BA produced" for "BAE" 
lower down. 






. h 

If AD, AE be the internal and external bisectors of the angle A in a. 
triangle of which the sides AB, AC are unequal, AC being the smaller, and 
if AD, AE meet BC and BC produced \n D, E respectively, 

the ratios of BD to i>Cand of BE to EC are alike equal to the ratio of 
BA to AC. 

Therefore BE is to ECtis BD to DC, 

that is, BE is to EC as the difference between BE and ED is to the 
difference between ED and EC, 

whence BE, ED, EC are in karmonit prograsion, or DE is a harmonic mtan 
between BE and EC, or again B, D, C, £ is a harm^nu range. 

Since the angle DAC is half of the angle BAC, 

and the angle CAE half of the angle CAF, 
while the angles BAC, CAF are equal to two right angles, 
the angle DAE is a right angle. 

Hence the circle described on DE as diameter passes through A. 

Now, if the ratio of BA to ^Cis given, and if BC is given, the points 
D, E on BC and BC produced are given, and therefore so is the circle on 
D, E as diameter. Hence M« /acus of a point sueh that its dittanas from two 
given points are in a given ratio {net being a ratio of equality) is a rirck. 

This locus was discussed by ApoUonius in his Plane Loci, Book ii., as we 
know fr«m Pappus (vii, p. 666), who says that the book contained the 
theorem that, if from two given jwjnts straight lines inflected to another 
point are in a given ratio, the point in which they meet will lie on either a 
straight line or a circumference of a circle. The straight line is of course the 
locus when the ratio is one of equality. The other case is quoted in the 
following form by Eutocius (ApoUonius, ed. Heiberg, ti. pp. \ 80—4). 

Given two points in a plane and a proportion between unequal strwght lines, 
it is possible to describe a circle in the plane so that the straight lines inflected 
from the given points to the dreumference of the circle shall have a ratio the 
tame as the given one. 

ApoUonius' construction, as given by Eutocius, is remarkable because he 
makes no use of either of the points D, E. He finds 0, the centre of the 
required circle, and the length of its radius directly from the data BC and the 
given ratio which we will call h : k. But the construction was not discovered 
by ApoUonius j it belongs to a much earlier date, since it appears in exactly 



VI. j] PROPOSITION 3 199 

the same rorm in Aristotle, Mekorolegica in. 5, J76 a 3 sqq. The 
analysis leading up to the construction is, as usual, not given either by 
Aristotle or Eutocius. We are told to take three straight lines x, CO (a 
length measured along BC produced beyond C, where J, C are the points at 
which the greater and smaller of the inflected lines respectively terminate), 
and r, such that, if h\khi the given ratio and h>k, 

k:h = hik + x, (a) 

■ '' x:SC=k:CO = h:r 08) 




This determines the position of O, and the length of r, the radius of the 
required circle. The circle is then drawn, any point P is taken on it and 
joined to B, C respectively, and it is proved that 

FB'.PC^h.k. 

We may conjecture that the analysis proceeded somewhat as follows. 

Ft would be seen that £, C are "conjugate points" with reference to the 
circle on DE as diameter. (Cf. ApoUonius, Cmks, i. 36, where it is proved, 
in terms, for a circle as well as for an ellipse and a hyperbola, that, if the 
polar of j5 meets the diameter DE in C, then EC: CD = EB : BD.) 

If O be the middle point of DE, and therefore the centre of the circle, 
D, E may be eliminated, as in the Conies, i. 37, thus. 

Since EC : CD = EB : BD, 

it follows that EC+CD: EC~ CD = EB ■¥ BD -.EB- BD, 

or iOD : 20C= tOB : tOD, '«■ ' 

that is, BO.OC= OD' = r', say. 

If therefore B be any point or the circle with centre O and radius r, 

BO: OP=OP:OC, 

so that 50/", PO Care similar triangles. ., .,, , ... 

\a.3A6:*:\on,h:k-BD:DC = BE-^EC ' ' •• 11 • • 

^BD + B£:D£ = BO:r. •• •'• ■-'"' 

Hence we require that 

BO:r = r:OC=BP:PC=h.k (8) 

Therefore, alternately, 

k:CO^h:r, 

which is the second relation in (^) above. '• ' 

Now assume a length x such that each of the last ratios is equal \ax\BC, 
as in (^). 



MiO BOOK VI [vi. 3, 4 

Then ' • . x:BC-k:CO = h:r. 

Therefore .r + A : BO -h:r, 

and, alternately, x + k:h = BO : r 

I =hik, from (S) above ; 

and this is the relation (a) which remained to be found. 

ApoUonius' proof of the construction is given by Eutocius, who begins by 
saying that it is manifest that r is a mean proportional between BO and OC. 
This IS seen as follows . 
From (j8) we derive 

x\BC=k: CO = A:r = {k-yx):BO, 
whence BO ■.r={k + x):h 

m:A:A, by <a), 
= r:CO, by(^), 
and therefore r* = BO . CO. 

But the triangles BOP, POC have the angle at common, and, since 
BO: OP = OF: OC, the triangles are similar and the angles OPC, OBP 
are equal 

[Up to this point Aristotle's proof is exactly the same ; from this point it 
diverges slightly.] 

If now CL be drawn parallel to BP meeting OP in L, the angles BPC 
ZCP are equal also. 

Therefore the triangles BPC, PCI. are similar, and 

BP:PC=PC:CL, 

whence BP^.PC^^BP: CL 

<= BO : OC, by parallels, 

= BO^ : OP* (sincere ; OP= OP: OC). 

Therefore BF:PC^BO:OP 

= A;*(for OP=-r). 

[Aristotle infers this more directly from the similar triangles P03, COP. 
Since these triangles are similar, 

OP: CP=OB:BP, 

whence BP: PC'' BO :0P "• 

= h:k.'\ 

ApoUonius proves lastly, by reductio ad ahsurdum, that the last equation 
cannot be true with reference to any point P which is not on the circle so 
described. 



Proposition 4. 

In equiangular triangles the sides about (he equal angles 
are proportional, and those are corresponding sides which 
subtend the equal angles. 




VI. 4] PROPOSITIONS 3, 4 sol 

Let ABC, DCE be equiangular triangles having the 
angle ABC equal to the angle 
DCE, the angle BAC to the 
angle CDE, and further the angle 
ACB to the angle CED ; 

I say that in the triangles ABC, 
DCE the sides about the equal 
angles are proportional, and those 
are corresponding sides which 
subtend the equal angles. 

For let BC be placed in a 
straight line with CE. 

Then, since the angles ABC, ACB are less than two right 
angles, [i, 17] 

and the angle ACB is equal to the angle DEC, 

therefore the angles ABC, DEC are less than two right 

angles ; 

therefore BA, ED, when produced, will meet. [i. Post. 5] 

Let them be produced and meet at E, 

Now, since the angle DCE is equal to the angle ABC, 

BF is parallel to CD. [1. *8] 

Again, since the angle ACB is equal to the angle DEC, 

AC is parallel to FE. [i. 38] 

Therefore FACD is a parallelogram ; 

therefore FA is equal to DC, and AC to FD. fi. 34] 

And, since AC has been drawn parallel to FE, one side 
of the triangle FBE, 

therefore, as BA is to AF, so is BC to CE. [vi. 1] 

But AF is equal to CD ; 

therefore, as BA is to CD, so is BC to CE, 

and alternately, as AB is to BC, so is DC to CE. [v. 16] 

'1 Again, since CD is parallel to BF, ' 

therefore, as BC is to CE, so is FD to DE. [vi. »] 

But FD is equal to AC; 

therefore, as BC is to CE ^o\^ AC to DE, 

and alternately, as BC is to CA, so is CE to ED. [v. r6] 



■M3 BOOK VI [VI. 4, s 

Since then it was proved that, ' 

as AB is to BC, so is DC to CE, 
and, as BC is to CA, so is CE to ED ; 

therefore, ex aeguali, as BA is to ^C, so is CD to DE. [v. sa] 

Therefore etc. 

Q, E. D, 

Todhunter remarks that " the manner in which the two triangles are to be 
placed is very imperfectly described; their bases are to be in the same straight 
line and contiguous, their vertices are to be on the same side of the base, and 
each of the two angles which have a common vertex is to be equal to the 
remote angle of the other triangle," But surely Euclid's description is 
sufficient, except for not saying that B and D must be on the same side 
of BCE. 

VI. 4 can be immediately deduced from vi. a if we superpose one triangle 
on the other three times in succession, so that each angle successively 
coincides with its equal, the triangles being similarly situated, e.g. if {A, B, C 
and D, E, F being the equal angles respectively) we apply the angle DEFio 
the angle ABC so that D lies on AB {produced if necessary) and J^on BC 
(produced if necessary). De Morgan prefers this method. " Abandon," he 
says, " the peculiar mode of construction by which Euclid proves two cases at 
once; make an angle coincide with its equal, and suppose this process repeated 
three times, one for each angle." 



. , Proposition 5. 

If two triangles have their sides proportional, the triangles 
will be equiangular and will have those angles equal which the 
corresponding sides subtend. 

Let ABC DEF be two triangles having their sides 
proportional, so that, 

as AB is to BC, so is DE to EF, 

sis BC is to C A, so is EF to FD, • ' ' 

and further, as BA is to AC, so is ED to DF; 

I say that the triangle ABC is equiangular with the triangle 
DEE, and they will have those angles equal which the corre- 
sponding sides subtend, namely the angle ABC to the angle 
DEE, the angle BCA to the angle EFD, and further the 
angle BAC to the angle EDF. 

For on the straight line EF, and at the points E, F on 
it, let there be constructed the angle FEG equal to the angle 
ABC, and the angle EFG equal to the angle A CB ; [i. aj] 



n. s] PROPOSITIONS 4. S Bp 

therefore the remaining angle at A is equal to the remaining 
angle at G. [t- 3*] 

Therefore the triangle ABC is equiangular with the 
triangle G£F. 





Therefore in the triangles ABC, GEF the sides about 
the equal angles are proportional, and those are corresponding 
sides which subtend the equal angles ; [vi. 4] 

therefore, as AB is to BC, so is GE to EF. 

But, as AB is to BC, so by hypothesis is DE to EF\ 

therefore, as DE is to EF, so is GE to EF. [v. n] 

Therefore each of the straight lines DE, GE has the 
same ratio to EF; 

therefore DE is equal to GE. [v. g] 

For the same reason 

DF is also equal to GF. 

Since then DE is equal to EG, 

and EF is common, 

the two sides DE, EF are equal to the two sides GE, EF; 

and the base DF is equal to the base EG ; 

therefore the angle DEF is equal to the angle GEF, [1. 8] 

and the triangle DEF is equal to the triangle GEF, 

and the remaining angles are equal to the remaining angles, 
namely those which the equal sides subtend. [1. 4] 

Therefore the angle DEE is also equal to the angle GFE, 

and the angle EDF to the angle EGF. 

And, since the angle FED is equal to the angle GEF, 
while the angle GEF is equal to the angle ABC, 
therefore the angle ABC is also equal 10 the angle DEF. 



:.aB4 BOOK VJ • [vi. s, 6 

For the same reason 

the angle ACB is also equal to the angle DFE^ "■■ 
and further, the angle at A to the angle at D ; 

therefore the triangle ABC is equiangular with the triangle 
DBF. 

Therefore etc. 

Q. E, D. 

This proposition is the complete converse, vi. 6 a partial converse, of vi. 4. 
Todhuntcr, after Walker, remarks that the enunciation should make it 
clear that the sides of the triangles laken in order are proportional. It is quite 
f»o&sible that there should be two triangles ABC, Z'£/'such that 

AB is to .SCas DE to EF, 
and .SCisto C^^as DF'xs, to EJ> (instead of ^i^to/Z)), 

so that A3 is to AC as J?Fto EF 

{fx aequali \n ptrlurbed prepartion)\ 

in this case the sides of the triangles are proportional, but not in the same 
order, and the triangles are not necessarily equiangular to one another. For a 
numerical illustration we may suppose the sides of one triangle to be 3, 4 and 
5 feet respectively, and those of another to be 11, 15 and 10 feet respectively. 
In VI. 5 there is the same apparent avoidance of indirect demonstration 
which has been noticed on t. 48. i •■. .m u 1 ■ t ' 



Proposition 6. 

If two triangles have one angle equal to one angle and the 
sides about the equal angles proportional, the triangles will be 
equiangular and will have those angles equal which the corre- 
sponding sides subtend. 

Let ABC, DEF be two triangles having one angle BAC 
equal to one angle EDF and the sides about the equal angles 
proportional, so that, 

as BA is to AC, so is ED to DF\ 
I say that the triangle ABC is equiangular with the triangle 
DEF, and will have the angle ABC equal to the angle DEF, 
and the angle ACB to the angle DFE. 

For on the straight line DF, and at the points D, Fon it, 
let there be constructed the angle FDG equal to either of the 
angles BAC, EDF, and the angle DFG equal to the angle 
ACB; [1.23] 

therefore the remaining angle at B is equal to the remaining 
angle at G, -. ... . , » .. [•• 3'] 



VI. 6J PROPOSITIONS s, 6 ^6$ 

Therefore the triangle ABC is equiangular with the 
triangle DGF. 

Therefore, proportionally, as BA is to AC, so is GD to 

DF. [vi. 4] 

But, by hypothesis, a.sBA is to AC, so also is £/? to I}F; 

therefore also, as FD is to /)F, so Is GD to DF. [v. n] 





Therefore £D is equal to BG ; • • ' [-^ ^j 

and Z?/^ is common ; 1 • 

therefore the two sides £D, DFare equal to the two sides 
GD, DF; and the angle EDF is equal to the angle GDF; 

therefore the base EF is equal to the base GF, ' 

and the triangle DEF is equal to the triangle DGF, 

and the remaining angles will be equal to the remaining angles, 
namely those which the equal sides subtend. [i. 4] 

Therefore the angle DFG is equal to the angle DFE, 
and the angle DGF to the angle DEF. 
But the angle DFG is equal to the angle ACB; ; 1 li-- 
therefore the angle ACB is also equal to the angle DFE. 

And, by hypothesis, the angle BAC is also equal to the 
angle EDF; 

therefore the remaining angle at B is also equal to the 
remaining angle at E\ ^ ■ y .1 ['-3*] 

therefore the triangle ABC is equiangular with the triangle 
DEF. 

Therefore etc. • < cj. ■ . ■' 

' ■ -' . ^ 1 ■ *• • ~ ' ;■> ■ Q. E, D. 




006 <•> ,t BOOK VI [Ti, 7 

Proposition 7. '•' •'•' ^'.f' ' 

1/ two triangles have one angle equal to one angle, the 
sides about other angles proportional, and th-e remaining angles 
either both less or both not less than a right angle, the triangles 
will be equiangular and will have those angles equal, the stdes 
about which are proportional. 

Let ABC, DEFhe two triangles having one angle equal 
to one angle, the angle BAC to 
the angle EJDF, the sides about 
other angles ABC, DEF propor- 
tional, so that, as AB ts to BC, 
so is DE to EF, and, first, each 
of the remaining angles at C, F 
less than a right angle ; 

I say that the triangle ABC is 

equiangular with the triangle 

DEF, the angle ABC will be 

equal to the angle DEF, and the remaining angle, namely 

the angle at C, equal to the remaining angle, the angle 

at F. 

For, if the angle ABC is unequal to the angle DEF, one 
of them is greater. 

Let the angle ABC be greater ; 

and on the straight line AB, and at the point B on it, let the 
angle ABG be constructed equal to the angle DEF. [i. 33] 

Then, since the angle A is -equal to D, a .:. ui 
and the angle ABG to the angle DEF, 

therefore the remaining angle A GB is equal to the remaining 
angle DFE. [i. jj] 

Therefore the triangle ABG is equiangular with the 
triangle DEF. 

Therefore, as AB is to BG, so is DE to EF [vi. 4] 

But, as DE is to EF, so by hypothesis is AB to BC\ 

therefore AB has the same ratio to each of the straight 
lines BC, BG ; [v. 11] 

therefore BC is equal to BG, [v. 9] 

so that the angle at C is also equal to the angle BGC. [i. 5] 



VL 7] PROPOSITION 7 307 

But, by hypothesis, the angle at C is less than a right 

angle ; 

therefore the angle BGC is also less than a right angle ; 

so that the angle A GB adjacent to it is greater than a right 
angle. [1. 13] 

And it was proved equal to the angle at F\ 
therefore the angle at F'\% also greater than a right angle. 

But it is by hypothesis less than a right angle : which is 
absurd. 

Therefore the angle ABC is not unequal to the angle 
DEF\ 

therefore it is equal to it. 

But the angle at A is also equal to the angle at D ; 

therefore the remaining angle at C is equal to the remaining 
angle at F, [i. 3a] 

Therefore the triangle ABC is equiangularwith the triangle 
DEF. 

But, again, let each of the angles at C, F be supposed not 
less than a right angle ; 
1 say again that, in this case too, the 
triangle ABC is equiangular with the 
triangle DEF. 

For, with the same construction, 
we can prove similarly that 

BC is equal to BG \ 

so that the angle at C is also equal to 
the angle BGC. \y s] 

But the angle at C is not less than a right angle ; 
therefore neither is the angle BGC less than a right angle. 

Thus in the triangle BGC the two angles are not less 
than two right angles : which is impossible, [i- 17] 

Therefore, once more, the angle ABC is not unequal to 
the angle DEF; 

therefore it is equal to it. 

But the angle at A is also equal to the angle at D ; 

therefore the remaining angle at C is equal to the remaining 
angle at F, [i, 33} 




Mft^ BOOK VI [vi. 7 

Therefore the triangle ABC is equiangular with the triangle 
DEF. 

Therefore etc. '■ . ■ ;- 

Q. E, D. 

Todhunter points out, after Walker, that some more words are necessary 
to make the enunciation precise : "If two triangles have one angle equal to one 
angle, the sides about other angles proportional <so that tht sides sttbttnding 
ihe equal anglts are homologous^. ..." 

This proposition is the extension to similar triangles of the ambiguous ^ase 
already mentioned as omitted by Euclid in relation to equality of triangles in 
all respects {cf. note following i. 26, Vol, 1. p. 306). The enunciation of vi. 7 
has suggested the ordinary method of enunciating the ambiguous (ase where 
equality and not similarity is in question. Cf. Todhunter's note on 1. 26, 

Another possible way of presenting this proposition is given by Todhunter, 
The essential theorem to prove is : 

ff two triangles have two sides of the one proportional to two sides of the 
other, and the angles opposite to one pair of corresponding sides equal, the angles 
which are opposite to the other pair of €orresponding sides shell either ^e equal or 
lie together equal to two right angles. 

For the angles included by the proportional sides must be either equal or 
uneqtiaL 

If they are equal, then, since the triangles have two angles of the one 
equal to two angles of the other, respectively, they are equiangular to one 
another. 

We have therefore only to consider the case in which the angles included 
by the proportional sides are unequal. 

The proof is, except at the end, like that of vi. 7. 

Let the triangles ABC, DEF have the angle at A equal to the angle at D ; 
let AB be to BC as DE to EF, 
but let the angle ABCht not equal to the angle DEF. 





The angles ACB^ DFE shall be together equal to two right angles. 

For one of the angles ABC, DEF must be the greater. 

Let ABC\^ the greater; and make the angle ABG equal to the angle 
DEF 

Then we prove, as in vi. 7, that the triangles ABG, DEF are equiangular, 
whence 

AB is to BG as DE is to EF. 

But AS is to BC as DE is to EF, by hypothesis. 

Therefore BG is equal to BC, 

and the angle BGC is equal to the angle BCA. 



VI. 7. 8] 



PROPOSITIONS 7, 8 



Now, since the triangles ABG, DMF'wfi equiangular, 

the angle SGA is equal to the angle EFD, 
Add to them respectively the equal angles BGC, EC A; therefore the 
angles BCA, EFD are together equal to the angles BGA, BGC, i.e. to two 

right angles. 

It follows therefore that the angles BCA, EFD must be either equal or 

supplementary. 

But (i), if each of them is less than a right angle, they cannot be 
supplementary, and they must therefore be equal ; 

(2) if each of them is greater than a right angle, Utey cannot be 
supplementary and must therefore be equal; 

(3) if one of them is a right angle, they are supplementary and also equal. 

Simson distinguishes the last case (3) in his enunciation : "then, if each of 
the remaining angles be either less or not less than a right angle, or if ont 0/ 
ihtm be a rigAt angle,., ," 

The change is right, on the principle of lestricring the conditions to the 
minimum necessary to enable the conclusion to be inferred. Simson adds a 
separate proof of the case in which one of the remaining angles is a right 
angle. 

" Lastly, let one of the angles at C, F, viz. the angle at C, be a right angle; 
in this case likewise the triangle ABC 
IS equiangular to the triangle DEF, 

For, if they be not equiangular, 
make, at the point B of the straight 
line AB, the ai^le ABG equal to the 
angle DEF\ then it may be proved, 
as in the first case, that BG vi equal 
vaBC. 

But the angle BCG is a right 
angle; 

therefore the angle BGC is also a 
right angle; 

whence two of the angles of the tri- 
angle BGC ^.Tc together not less than 
two right angles : which is impossible. 
Therefore the triangle ABC is equiangular to the triangle DEF." 

Proposition 8, . , 

1/ in a rigkt-anghd triangle a perpendicular be drawn 
from ike right angle to the base, ike triangles adjoining ike 
perpendicular are similar both to the whole and to one another. 

Let ABC be a right-angled triangle having the angle 
BAC right, and let AD be drawn from A perpendicular 
to^C; 

I say that each of the triangles ABD, ADC is similar to 
the whole ABC and, further, they are similar to one another. 





ue BOOK VI [vi. 8 

For, since the angle BAC is equal to the angle ADB, 
for each is right, 

and the angle at B is common to the 
two triangles ABC and ABD, 
therefore the remaining angle ACB 
is equal to the remaining angle 
BAD ; [.. 3^] 

therefore the triangle ABC is equi- 
angular with the triangle ABD. 

Therefore, as BC which subtends the right angle in the 
triangle ABC is to BA which subtends the right angle in 
the triangle ABD, ^o\% AB itself which subtends the angle 
at C in the triangle ABC to BD which subtends the equal 
angle BAD in the triangle ABD, and so also '\% AC to AD 
which subtends the angle at B common to the two triangles. 

[v,.4] 

Therefore the triangle ABC is both equiangular to the 
triangle ABD and has the sides about the equal angles 
proportional. 

Therefore the triangle ABC is similar to the triangle 
ABD. [VI. Def. i] 

Similarly we can prove that 
the triangle ABC is also similar to the triangle ADC ; 
therefore each of the triangles ABD, ADC is similar to the 
whole ABC. 

I say next that the triangles ABD, ADC are also similar 
to one another. 

For, since the right angle BDA is equal to the right angle 
ADC, 

and moreover the angle BAD was also proved equal to the 
angle at C, 

therefore the remaining angle at B is also equal to the 
remaining angle DAC; [»• 32] 

therefore the triangle ABD is equiangular with the triangle 
ADC. 

Therefore, as BD which subtends the angle BAD in the 
triangle ABD is to DA which subtends the angle at C in the 
triangle ADC equal to the angle BAD, so is AD itself 
which subtends the angle at B in the triangle ABD to DC 
which subtends the angle DAC in the triangle ADC equal 



VI. 8, g] PROPOSITIONS 8, 9 * jii 

to the angle at B, and so also is BA to AC, these sides 
subtending the right angles ; [vi. 4] 

therefore the triangle ABD is similar to the triangle ADC. 

[vi. Def. i] 
Therefore etc. 

PoRiSM. From this it is clear that, if in a right-angled 
triangle a perpendicular be drawn from the right angle to the 
base, the straight Hne so drawn is a mean proportional 
between the segments of the base. q. e. d. 

Sim son remarks on this proposition : "It seems plain that some editor 
has changed the demonstration that Euclid gave of this proposition : For, 
after he has demonstrated that the triangles are equiangular to one another, 
he particularly shows that their sides about the equal angles are proportionals, 
as if this had not been done in the demonstration of prop, 4 of this book : 
this superfluous part is not found in the translation from the Arabic, and is 
now left out." 

This seems a little hypercritical, for the "particular showing" that the 
sides about the e<jual angles are proportionals is really nothing more than 
a somewhat full citation of vi. 4. Moreover to shorten his proof still 
morci Simson says, after proving that each of the triangles ABD, ADC is 
similar to the whole triangle ABC, "And the triangles ABD, ADC being 
both equiangular and similar to ABC are equiangular and similar to one 
another," thus assuming a particular case of vi. zi, which might well be 
proved here, as EucSid proves it, with somewhat more detail. 

We observe that, here as generally, Euclid seems to disdain to give the 
reader such small help as might be afforded by arranging the letters used to 
denote the triangles so as to show the corresponding angular points in the 
same order for each pair of triangles ; A is the first letter throughout, and the 
other two for each triangle are in the order of the figure from left Co righL It 
may be in compensation for this that he states at such length which side 
corresponds to which when he comes to the proportions. 

In the Greek texts there is an addition to the Pori^m inserted after 
"(Being) what it iras required to prove," viz. "and further that between the 
base and any one of the segments the side adjacent to the s^ment is a mean 
proportional" Heiberg concludes that these words are an interpolation 
(i) because they come after the words wrcp iSh Stifm which as a rule follow the 
Porism, (2) they are absent from the best Theonine MSS., though P and 
Campanus have them without the wip {£« Stufiu. Heiberg's view seetns to 
be confirmed by the fact noted by Austin, that, whereas the first part of the 
Porism is quoted later in vi. 15, in the lemma before x. 33 and in the lemma 
after xin. 13, the second part \& prmted vci the former lemma, and elsewhere, 
as also in Pappus (in. p. 72, 9— »3). 

Proposition 9, 

From a given straight line to cut off a prescribed part. 
Let AB be the given straight line ; 
thus it is required to cut off from AB a prescribed part 



lis 



BOOK VI 



[«.» 



Let the third part be that prescribed. 
S Let a straight line AC he drawn through from A con- 
taining with AS any angle ; 

let a point /? be taken at random on 
AC, and let DB, EC be made equal 
to AD. [1- 3] 




10 Let ^Cbe joined, and through D 
let DF be drawn parallel to it. [i- 31] 
Then, since FD has been drawn 
parallel to BC, one of the sides of the triangle ABC, 

therefore, proportionally, as CD is to DA, so is BFto FA. 

1 • [n. a] 

•S But CD is double of DA ; 

therefore BF is also double of FA ; - 

therefore BA is triple of AF. 

Therefore from the given straight line AB the prescribed 
third part AF has been cut off. 

Q, E, F. 

6. any angle. The exptessian here and in the two foltowing propositions is ruxoura 
yurtu, corresponding exactly to Tirj^ir (Ttj^miw which I have Iransfated u "« point (taken) 
at randffm"^ but *'an angle (talten) at random" would not be so appropriate where it is a 
question, not of taking any angle at all, but of drawing a straight line casuAlly so as to make 
any angle with another straight line. 

Simson observes that " this b demonstrated in a parti cuUt case, viz. that 
in which the third part of a straight line is required to be cut off; which b 
not at all like Euclid's inanner. Besides, the author of that demonstration, 
from four magnitudes being proportionals, concludes that the third of them is 
the same multiple of the fourth which the first is of the second ; now this is 
nowhere demonstrated in the sth book, as we now have it ; but the editor 
assumes it from the confused notion which the vulgar have of proportionals." 

The truth of the assumption referred to is proved by Simson in hb 
proposition D given above (p, laS); hence he is 
able to supply a general and legitimate proof 
of the present proposition. A 

" I^et AB be the given straight line ; it b 
required to cut oW any part from it. 

From the point A draw a straight line AC 
making any angle with AB; in AC take any 
point 2>, and take ^4 C the same multiple of AD 
that AB is of the part which is to be cut off 
from it ; 

join SC, and draw jD£ parallel to it ; 

then A Eh the part required to be cut off. 




VI. 9, lo] PROPOSITIONS 9, 10 "3 

Because ED is parallel to one of the sides of the triangle ABC, \'a, to BC, 
as CD is to DA, so is BE to EA, [vi. 2] 

and, eomponaido, 

CA is to AD, as S A to AE. [v. 1 8] 

But CA is a multiple of AD ; 
therefore BA is the same multiple of AE. [Prop. D] 

Whatever part therefore AD is of AC, AE is the same part of AB ; 
wherefore from the straight line AB the part required is cut off." 

The use of Simson's Prop. D can be avoided, as noted by Camerer after 
Baermann, in the following way. We first prove, as above, that 
CA is to AD as BA is to AE. 
Then we infer that, alternately, 

CA is to BA as AD to AE. [v. 16] 

But AD is to AE as n . AD to n . AE 

{where n is the number of times that AD\% contained m AC); [v, 15] 

whence ACkIo AB as n . AD is to « . AE. [v, 1 1] 

In this proportion the first term is equal to the third ; therefore [v. 14] 

the second is equal to the fourth, 

so that AB is equal to n times AE. 
Prop. 9 is of course only a particular case of Prop 10. ' " 

Proposition id. 

To cut a given uncut straight line similarly to a given cut 
straight line. 

Let AB be the given uncut straight line, and AC the 
straight line cut at the points D, 
E ; and let them be so placed as 
to contain any angle ; 
let CB be joined, and through D, 
E let DF, EG be drawn parallel 
to BC, and through D let DHK 
be drawn parallel to AB. [1, 31] 

Therefore each of the figures 
FH, HB is a parallelogram ; 
therefore DH is equal to FG and HK to GB. [i. 34] 

Now, since the straight line HE has been drawn parallel 
to KC, one of the sides of the triangle DKC, 

therefore, proportionally, as CE is to ED, so is KH to HD. 

[vi. j] 




i«»4 "' BOOK VI [vi. 10, II 

^ But KH is equal to BG, and HD to GF; ■ ' 

therefore, as C£ is to ED, so is 3G to GF, 

Again, since FD has been drawn parallel to GF, one ot 
the sides of the triangle j4 GF, 
therefore, proportionally, as FD is to DA, so is GF to FA. 

[vt. j] 

But it was also proved that, 

as CF is to FD, so is FG to GF; 
therefore, as CF is to FD, so is BG to GF, ^ 

and, as FD is to /?^, so is GF to /^^. 
Therefore the given uncut straight line AB has been cut 
similarly to the given cut straight line AC. 

Q. E. F. 

Proposition ii. 

' - ' To two given straight lines to find a third proportional. 

Let BA, AC be the two given straight lines, and let 
them be placed so as to contain any 
angle ; 

thus it is required to find a third pro- 
portional to BA, AC. 

For let them be produced to the 
points D, E, and let BD be made equal 
to^C; [t. 3] 

let BC be joined, and through D let DF 
be drawn parallel to it. [i. 31] 

Since, then, BC has been drawn 
parallel to DF, one of the sides of the triangle ADE, 
proportionally, as AB is to BD, so is ^C to CF. [vi. aj 

But BD is equal to AC; 
therefore, as AB is to AC, so is AC tc CE. 

Therefore to two given straight lines AB, AC 3. third 

proportional to them, CF, has been found, 

Q. E. F. 

I. to And. The Greek word, bat and in tbe Bext two piopoHttom, is ir pnirivptir, 
liter*] I7 "to find in addilien." 

This proposition is again a particular case of the succeeding Prop, i », 
Given a ratio between straight lines, VI, ii enables us to find the ratio 
which i$ its duplicate;. 




vt. ii] PROPOSITIONS io-~u 1^ 

t. ,. 

Propos:tion 12. 
To three given straight lines to find a fourth proportional. 
Let A, B, C be the three given straight lines ; 
thus it is required to find a fourth proportional to A, B, C. 




B- 

C- 



Let two straight lines DE, DF be set out containing any 
angle EDF ; 

let DG be made equal to A, GE equal to B, and further DH 
equal to C\ 

let GH be joined, and let EF be drawn through E parallel 
to it. [1. 31] 

Since, then, GH has been drawn parallel to EF, one of 
the sides of the triangle DEF, 
therefore, as DG is to GE, so is DH to HF. [vi. a] 

But DG is equal to A, GE to B, and DH to C ; 
therefore, as A is to B, so is C to HF. 

Therefore to the three given straight lines A,B,Ca. fourth 
proportional HF has been found. 

Q. E..F. 

We have here the geometrical equivalent of the " rule of three." 

It is of course immaterial whether, as iti Euclid's proof, the first and 

second straight lines are measured on one of the lines forming the angle and 

the third on the other, or the first and third are measured on one and the 

second on the other. 

If it should be desired that the first and the required fourth be measured 

on one of the lines, and the second and third on 

the other, we can use the following construction. 

Measure -DE on one straight line equal to A, and 

on any other straight line making an angle with 

the first at the point D measure I>F equal to £, 

and DG equal to C, Join B.F, and through G 

draw GJf an/i-fiamiltl to EF, le. make the angle 

DGH equal to the angle DEF; let GH meet 

DE (produced if necessary) in H. 





»« BOOK VI [vi, I a— 14 

DHis then the fourth proportional. 

For the triangles EDF, GDH are similar, and the sides about the equal 
angles are proportional, so that 

DE is to DFa& DG to DH, 

OT ^ is to ^ as C to DJf, 



Proposition 13. 
To two given straight lines to find a mean proportional. 

Let AB, BC be the two given straight lines ; 
thus it is required to find a mean 
proportional to AB, BC. 

Let them be placed in a straight 
line, and let the semicircle ADC be 
described or\ AC \ 

let BD be drawn from the point B at 
right angles to the straight line AC, , 

and let AD, DC be joined. 

Since the angle ADC is an angle in a semicircle, it is 
right. [ill. 31] 

And, since, in the right-angled triangle ADC, DB has 
been drawn from the right angle perpendicular to the base, 
therefore DB is a mean proportional between the segments of 
the base, AB, BC. [vi. 8, For.] 

Therefore to the two given straight lines AB^ BC a mean 
proportional DB has been found, 

Q. E. F. 

This proposition, the Book vi. version of ii. 14, is equivalent to the 
extraction of the square root. It further enables us, given a ratio between 
straight lines, to find the ratio which is its sub-dupiieate, or the ratio of which 
it is duplicate. , - 



Proposition 14, ', ^ 

In equal and equiangular parallelograms the sides about 
the equal angles are reciprocally proportional ; and equiangular 
parallelograms in which the sides about (he equal angles are 
reciprocally proportional are equal 




vj. 14] PROPOSITIONS 12— 1 4 at? 

■^•^ Let AB, BC be equal and equiangular parallelograms 
having the angles at B equal, and 
let DB, BE be placed in a straight 
line ; 

therefore FB, BG are also in 
a straight line. [i. m] 

I say that, in AB, BC, the 
sides about the equal angles are 
reciprocally proportional, that is to 
say, that, as DB is to BE, so is 
GB to BF. 

For let the parallelogram FE be completed. 

Since, then, the parallelogram AB is equal to the parallelo- 
gram BC-, 

and FE is another area, 
therefore, as AB is to FE, so is BC to FE. [v. 7] 

But, as W^ is to FE, so is DB to BE, [vi, i] 

and, as BC is to FE, so is GB to BF, \id^ 

therefore also, as DB is to BE, so is GB to BF. [v. n] 

Therefore in the parallelograms AB, BC the sides about 
the equal angles are reciprocally proportional. 

,^i Next, let GB be to BF as DB to BE-, .n 

I say that the parallelogram AB is equal to the parallelogram 
BC. 

For since, as DB Is to BE, so is GB to BF, 
while, as DB is to BE, so is the parallelogram AB to the 
parallelogram FE, [vi. i] 

and, as GB is to BF, so is the parallelogram BC to the 
parallelogram FE, [vi. i] 

therefore also, as AB is to FE, so is ^C to FE ; [v. n] 

therefore the parallelogram AB is equal to the parallelogram 
BC ... ^ ,. . . [V.9] 

Therefore etc. 

Q. E. D. 

De Morgan says upon this proposition : " Owing to the disjointed manner 
in which Euctid treats compound ratio, this prrvpositton is strangely out of 
place. It is a particular case of vi, 23, being that in which the ratio of the 
sides, compounded, gives a ratio of equality. The proper definition of four 
mapiitudes being reciprocally proportional is that the ratio compounded of 
thetr ratios is that of equality." 



3i8 BOOK VI [vi. 14 

It IS true that vi. 14 is a particular /case of vi. 23, but, if either is out of 
platt, it is rather the latter that should be placed before vi. 14, since most of 
the propositions between vi, 15 and vi. 23 depend upon vi. 14 and 15. But 
is perfectly consistent with Euclid's manner to give a particular case first 
and its extension later, and such an arrangement often has great advantages 
in that it enables the more difficult parts of a subject to be led up to more 
easily and gradually. Now, if De Morgan's view were here followed, we 
should, as it seems to me, be committing the mistake of explaining what is 
relatively easy to understand, viz, two ratios of which one is the inverse of 
the other, by a more complicated conception, that of compound ratio. In 
other words, it is easier for a learner to realise the relation indicated by the 
statement that the sides of equal and equiangular parallelograms are "recipro- 
cally proportior^al " than to form a conception of parallelograms such that 
" the ratio compounded of the ratio of their sides is one of equality." For 
this' reason I would adhere to Euclid's arrangement. 

The conclusion that, since I>B, BE are placed in a straight line, ES, BG 
are also in a straight line is referred to t. 14. The deduction is made clearer 
by the following steps. 

The angle DBF'm equal to the angle GBE; " 

add to each the angle FB£ ; 

therefore the angles DBF, FBE are together equal to the angles GBE, FBE. 

{C. N. i] 

But the angles DBF, FBE are together equal to two right angles, [i. 13] 

therefore the angles GBE, FBE are together equal to two right angles, 

[C.N.t] 

and hence FB, BG are in one straight line. [i. 14] 

The result is also obvious from the converse of 1. 15 given by Proclus 
(see note on i. 15). 

The proposition vi. 14 contains a theorem and one partial converse of it; 
so also does vi. 1 5. To each proposition may be added the other partial 
converse, which may be enunciated as follows, the words in square brackets 
applying to the case of triangles (vi. 15). 





Equal paralhlogrami \triattgUs\ which havt the sides absut one angle in 
each redprocally proportional art equiansular \have the angles included by those 
sides either equal or supplementary^ 

Let ABf BC be equal parallelograms, or let FBD, EBG be equal 



■n. 14, isl PROPOSITIONS 14, 15 119 

triangles, such that the sides about the angles at B are reciprocally propor- 
tional, i.e. such that 

DB : BE = GB : BF. 

We shall prove that the angles FED, EBG are either equal or supple- 
mentary. 

Place the figures so that DB. BE are in one straight line. 

Then FB, BG are either in a straight line, or not in a straight line. 

(i) If FB, BG are in a straight line, the figure of the proposition 
(with the diagonals FD, EG drawn) represents the facts, and 

the angle FBD is equal to the angle EBG. ['. 15] 

(2) If J'B, BG are not in a straight line, 
produce FB to H so that BH may be equal to BG. 

Join EJf, and complete the parallelogram EBHK. 

Now, since DB : BE ^ GB \ BF 

xbA. GB = HB, ~ 

DB : BE = HB . BF, 
and therefore, by vi. r4 or 15, 
the parallelograms j4B, BK 3.x^ equal, or the triangles FBD, EB/fare equal. 

But the parallelograms AB, BCart eciual, and the triangles FBD, EBG 
are equal ; 

therefore the parallelograms BC, BK are equal, and the triangles EBH, 
EBG are equal. 

Therefore these parallelograms or triangles are within the same parallels : 
that is, G, C, H, K are in a straight line which is parallel to DE. [1, 39] 

Now, since BG, BHare. equal, 
the angles BGH, BHG are equal. 

By parallels, it follows that 

the angle EBG is equa' to the angle DBH, 
whence the angle EBG is supplementary to the angle FBD. 



Proposition 15. 

In equal triangles which have one angle equal to one angle 
the sides about the equal angles are reciprocally proportional ; 
and those triangles which have otie angle equal to one angle, 
and in which the sides about the equal angles are reciprocally 
proportional, are equal 

Let ABC, ADE be equal triangles having one angle 
equal to one angle, namely the angle BAC to the angle 
DAE, 

I say that in the triangles ABC, ADE the sides about the 

equal angles are reciprocally proportional, that is to say, that, 

as CA is to AD, so is EA to AB, 




aao BOOK VI [vi. 15 

For let them be placed so that CA is in a straight 
line with AD; 

therefore EA is also in a straight line with 
AB. [i. 14] 

Let BD be joined. 

Since then the triangle ABC is equal to 
the triangle ADE, and BAD is another 
area, 

therefore, as the triangle CAB is to the 
triangle BAD, so is the triangle EAD to 
the triangle BAD. [v. 7] 

But, as CAB is to BAD, so is CA to AD, [vi. i] 

and, as EAD is to BAD, so is EA to AB. \id:\ 

Therefore also, as CA is to AD, so is EA to AB. (v. n] 
Therefore in the triangles ABC, ADE the sides about 
the equal angles are reciprocally proportional. 

Next, let the sides of the triangles ABC, ADE be reci- 
procally proportional, that is to say, let EA be to AB as CA 
to AD ; 

I say that the triangle ABC is equal to the triangle ADE. 

For, if BD be again joined, 1 , 

since, as CA is to AD, so is EA to AB, 

while, as CA is to AD, so is the triangle ABC to the triangle 
BAD, 

and, as EA is to AB, so is the triangle EAD to the triangle 
BAD, [vt. ,] 

therefore, as the triangle ABC is to the triangle BAD, so is 
the triangle EAD to the triangle BAD. [v. 11] 

Therefore each of the triangles ABC, EAD has the same 
ratio to BAD. 

Therefore the triangle ABC is equal to the triangle EAD. 

[v. 9] 
Therefore etc. ■ -> ;■ r 

Q. E. D. 

As indicated in the partial converse given in the last note, this proposition 
is equally true if the angle included by the two sides in one triangle h 
supplementary, instead of being equal, to the angle included by the two sides 
in the other. 




VI. 15, 1 6] PROPOSITIONS 15, i« »ai 

Let ABC, ADE be two tria.ngles such that the angles BA C, DAE are 
supplementary, and also .^ 

CA:AD = EA: AB. 

In this case we can place the triangles so that 
CA is in a straight line with AD, and AB lies 
along AE (since the angle EAC, being supple- 
mentary to the angle EAD, is equal to the anglo 
BAC). 

If we join BD, the proof given by Euclid 
applies to this case also. 

It is true that vi, 15 can be immediately inferred from vi. 14, since a 
triangle is half of a parallelogram vrith the same base and height. But, 
Euclid's object being to give the student a grasp of mttkods rather than 
results, there seems to be no advantage in deducing one proposition from the 
other instead of using the same method on each. 



Proposition 16. 

If four straight lines be proportional, the rectangle con- 
tained by the extremes is equal to the rectangle contained by 
the means ; and, if the rectangle contained by the extremes be 
equal to the rectangle contained by the means, the four straight 
lines will be proportional. 

Let the four straight lines AB, CD, E, F be propoitional, 
so that, as AB is to CD, so is B to F; 

I say that the rectangle contained by AB, F is equal to the 
rectangle contained by CD, E. 



Let AG, CH be drawn from the points A, C ^t right 
angles to the straight lines AB, CD, and let AG he made 
equal to F, and Cff equal to E. 

Let the parallelograms BG, DH be completed. 

Then since, as AB Is to CD, so is .£■ to F, 
while E is equal to CH, and f^ to AG, 
therefore, as AB is to CD, so is CH to AG. 

Therefore in the parallelograms BG, DH the sides about 
the equal angles are reciprocally proportional. 



3«3 BOOK VI . , ■ [VI. 1 6 

But those equiangular parallelograms in which the sides 
about the equal angles are reciprocally proportional are equal ; 

[VI. 14] 
therefore the parallelogram BG is equal to the parallelogram 

And BG is the rectangle AB, F, for AG is equal to F; 
and DH is the rectangle CD, E, for ^ is equal to CH \ 
therefore the rectangle contained by AB, F is equal to the 
rectangle contained by CD, E. 

Next, let the rectangle contained by AB, F be equal to 
the rectangle contained by CD, E ; 

I say that the four straight lines will be proportional, so that, 
as AB is to CD, so is E to F. 

For, with the same construction, 
since the rectangle AB, /^ is equal to the rectangle CD, E, 
and the rectangle AB, F is BG, for AG is equal to F, 
and the rectangle CD, £ is D/f, for C// is equal to £, 
therefore BG is equal to D/f. 

And they are equiangular , •, •• ■ . 

But in equal and equiangular parallelograms the sides about 
the equal angles are reciprocally proportional. [vi. 14] 

Therefore, as AB is to CD, so is Cff to A G. 

But CH is equal to E, and AG to F; 
therefore, as AB is to CD, so is £" to F. 

Therefore etc. q. e. d. 

Thb proposition is a, particular case of vi. 14, but one which is on all 
accounts worth separate statement. It may also be enuticiated in the follow- 
ing form : 

Jitetartgki which have thtir bases rtdprocaUy proporiumal to ihcir Atighis- 
art equcd in arta; and egml rectangles have their bases redproealfy proportional 
to thdr heights. 

Since any fkarallelogTam is equal to a rectangle of the same height and 
Dn the same base, and any triangle with the same height and on the same 
base is equal to half the parallelc^ram or rectangle, it follows that Equal 
parallelograms or triangles have their bases reciprocally proportional to their 
heights and vice vena. 

The present place is suitable for giving certain important propositions, 
including those which Simson adds to Book vi. as Props, B, C and D, which 
are proved directly by means of vi, 16. 

I. Proposition B is a particular case of the following theorem. 
J/ a circle be circumscribed about a triangle ABC and there be drawn through 
A any two straight lines either both within or both without the an^ BAG, pm. 



VI. 1 6] 



PROPOSITION r6 



'«t3 



AD mteting BC {produad if Mcttsary) in D and AE mealing the circle again 
in E, suth that the angles DAB, EAC are equal, then the rectangle AD, AE is 
equal to the rectangle BA, AC. 





Join CE. 

The angles BAD, £AC 3,k equal, by hypothesis ; 

and the angles ABZ>, AEC are equal, [in. n, as] 

Therefore the triangles ABD, AEC are equiangular. 

Hence BA is to AD as EA is to AC, 

and therefore the rectangle BA, AC is equal to the rectangle AD, AE, 

[VI. 1 6] 
There are now two particular cases to be considered. 

(a) Suppose that AD, AE coincide ; 

ADE will then bisect the angle BAC. ' • 

(b) Suppose that AD, v4£ are in one straight line but that D, E ate on 
opposite sides of A ; 

AD will then bisect the external angle at A. 

e 





In the fiist case {a) we have a; >•' - < 

the rectangle BA, ^iC equal to the rectangle EA, AD; 

and the rectangle EA, AD is equal to the rectangle ED, DA together with 
the square on AD, [\i. 3] 

i.e. to the rectangle BD, DC together with the square on AD. [11 1. 35] 

Therefore the rectangle BA, AC b, equal to the rectangle BD, DC 
tf^etber with the square on AD. [This is Simson's Prop B] 

In case (^) the rectangle EA, AD is equal to the excess of the rectangle 
ED, DA over the square on AD ; 

therefore the rectangle BA, AC k equal to the excess of the rectan^e BD, 
DC over the square on AD. 



a$4 BOOK VI [vt. i6 

The following converse of Simson's Prop. B may be given : ^ a stra^ht 
Uiu AD ^ drawn from the virtex K of a Iriat^lt to meH the base, S9 that the 
square on AD together with the recian^e BD, DC is tqual to the rectangU BA, 
AC, the line AD will biieet the angle BAG exapt when the sides AB, AC are 
tqual, in which atse every line drawn to the base will have the property men- 
timed, 

Let the circumscribed circle be drawn, and let AD produced meet it in 
E; join CE. 

The rectangle S£>, DC is equal to the rectangle ED, DA. [iii. 35] 

Add to each the square on AD ; 
therefore the rectangle BA, AC\s equal to the rectangle EA, AD. 

[hyp. and Ii. 3] 

Hence AS is to AD as AE to AC. [vi, 16] 

But the angle ABD is equal to the angle AEC. [111. 11] 

Therefore the angles BDA, EC A are either equal or supplementary. 

[vi. 7 and note] 

(o) If they are equal, the angles BAD, SAC 
are also equal, and AD bisects the angle BAC. 

{b) If they are supplementary, the angle ADC 
must be equal to the angle ACE. 

Therefore the angles BAD, ABD are together 
equal to the angles ACB, BCE, i.e. to the angles 
A CD, BAD. 

Take away the common angle BAD, and 
the angles ABD, ACD are equal, or 
jJJ is equal to AC. 

Euclid himself assumes, in Prop. 67 of the Data, the result of so much of 
this proposition as relates to the case where BA = AC. He assumes namely, 
without proof, that, if BA -^AC, and if ZJ be any point on BC, the rectangle 
BD, Z>C together with the square on AD\^ equal to the square on AB. 

Proposition C. 

Jffrom any angle of a triangle a straight line be drawn perpendicular to the 
opposite side, the rectangle contained by the other two sides of the triangle is equal 
to the rectangle contained by the perpendicular and the diameter of the circle 
(iraimscriied about the triangle. 

Let .i4^C be a triangle and AD the perpendicular on AB. Draw the 
diameter A£ of the circle circumscribed about the triangle ABC. 






Then shall the rectangle BA, ACh^ equal to the rectangle EA, AD. 
Join EC. 



VI. 1 6] 



PROPOSITION 1 6 



"5 



Since the right angle BDA is equal to the right angle ECA in a semi- 
circle, [ill. 31] 
and the angles ABD, AEC in the same s^pnent are equal, [in. 21] 
the triangles ABD, AEC are equiangular. 

Therefore, as ^^4 is to AD, so is EA to AC, [vj. 4] 

whence the rectangle BA, AC is equal to the rectangle EA, AD. [vi, 16] 

This result corresponds to the trigonometrical formula, for JP, the radius of 
the circumscribed circle, 



E = 



4& 



Proposition D. 

This is the highly important lemma given by Ptolemy (ed, Heiberg, Vol. i, 
pp. 36 — 7) which is the basis of his calculation of the table of chords in the 
section of Book i. of (he /xcyoAi; trvrraii% entitled " concerning the siite of the 
straight lines [i.e. chords] in the circle " (rtpl i-ijt TnjKiKonfTot too' ^v riji KixXif 

The theorem may be enunciated thus. 

7%e ruiangk wfitaimd by iht diagsnah of any quadrilateral inscribtdin a 
drclt is equal to the sum afthe nclangks contained by the pairs of opposite sides. 

I shall give the proof in Ptolemy's words, with the addition only, in 
brackets, of two words applying to a second figure not giver by Ptolemy. 

" Let there be a circle with any quadrilateral ABCD inscribed in it, and 
\exAC, BD be joined 

It is to be proved that the rectangle contained by ACznA BD is equal 
to the sum of the rectangles AB, DC and AD, BC. 

For let the angle ABE be made equal to the angle contained by DB, BC. 





-i' (jMl 

If then we add [or subtract] the angle EBD, ' ~ < 

the angle ABD will also be equal to the angle ESC. 

But the angle BDA is also equal to the angle BCE, ' [m. ai] 

for they subtend the same segment ; 
therefore the triangle ABD is equiangular with the triangle EBC. 

Hence, proportionally, 

as BC is to CE, so is BD to DA. [vi. 4] 

Therefore the rectangle BC,ADk equal to the rectangle BD, CE. 

[VI. 16] 
Again, since the angle ABE is equal to the angle DBC, 
and the angle BAE is also equal to the angle BDC, [m. ai) 

the triangle ABE is equiangular with the triangle DBC. 



aa6 



BOOK VI 



[vi. 1 6 



[VI. 4] 
[vi. i6] 



Therefore, proportionallj', 

as £A is to A£, so is SJD to DC; 
therefore the rectangle BA, DC is equal to the rectangle BD, AE. 

But it was also proved that 

the rectangle BC, AD is equal to the rectangle BD^ CE\ 
therefore the rectangle AC, BD as a whole is equal to the sum of the 
rectangles AB, i»Cand AD, BC: 

(being) what it was required to prove." 

Another proof of this proposition, and of its converse, is indicated \yj 
Dr Lachlan (Elements of Euclid, pp. 273 — 4). It depends on two preliminary 
propositions. 

(1) If two tirdes bi divided, by a chord in each, into segments which are 
similar respeettvcfy, the chords are proportional to the corresponding diameters. 

The proof is instantaneous if we join the ends of each chord to the centre 
of the circle which it divides, when we obtain two similar triangles. 

{1) IfYibe any point on the circle circumscribed about a triangle ABC, and 
DX, DY, DZ be perpendicular to the sides BC, CA, AB of the triangle 
respectively, then X, Y, Z lie in one straight line ; and, conversely, if the feet of 
the perpendiculars from any point D on the sides of a triangle lie in one straight 
line, D ties on the circle circumscribed about the triangle. 

The proof depending on ill. 21, 22 is well known. 

Now suppose that D is any point in the plane of a triangle ABC, and 
that DX, D y, DZ are perpendicular to the sides 
BC, CA, AB respectively. 

Join YZ, DA. 

Then, since the angles at ]^ Z are right, 
A, Y, D, Z lie on a circle of which DA is the 
diameter. 

And YZ divides this circle into segments which 
are similar respectively to the segments into which 
BC divides the circle circumscribing ABC, since 
the angles ZAY, BAC coincide, and their supple- 
ments are equal. 

Therefore, if i^ be the diameter of the circle 
circumscribing ABC, 

BChlodas YZistoDA; 
and therefore the rectangle AD, BC is equal to the rectangle d, YZ. 

Similariy the rettangle BD, CA is equal to the rectangle d, ZX, and the 
rectangle CD, AB\& equal to the rectangle d, XY. 

Hence, in a quadrilateral in general, the rectangle 
contained by the diagonals is less than the sum of the 
rectangles contained by the pairs of opposite sides. 

Next, suppose that D lies on the circle circum- 
scribed about ABC, but so that A, B, C, D follow 
each other on the circle in this order, as in the figure 
annexed. 

Let DX, DY, DZ be perpendicular to BC, CA, 
AB respectively, so that X, Y, Zare in a straight line. 

Then, since the rectangles AD, BC; BD, CA; CD, AB are equal to the 
rectangles d, YZ; d, ZX; d, .VF respectively, and XZi% equal to the sum of 





VI. i6] PROPOSITION id aaf 

Xy, YZ, so that the rectangle 4 XZ is equal to the sum of the rectangles 
4 JfKand d, YZ, It follows that 

the rectangle AC, BD is equal to the sum of the rectangles AD, BCand 
AB, CD. 

Cenvirsefy, if the latter statement is true, while we are supposed to know 
nothing about the position of D, it follows that 

XZ must be equal to the sum of XY, YZ, 
so that X, Y, Z must be in a straight line. 

Hence, from the theorem (2) above, it follows that D must lie on the 
circle circumscribed about ABC, i.e. that A BCD is a quadrilateral about 
which a circle can be described. 

All the above propositions can be proved on the basis of Book iii. and 
without using Book vi., since it is possible by the aid of 111.^21 and 35 alone 
to prove that in equiaitgular triangles the rectangles contained by the iwh- 
correspsnding sides about equal angles are equal to one another (a result arrived 
at by combining vi, 4 and v[. 16). This is the method adopted by Casey, 
H. M. Taylor, and Lachlan ; but I fail to see any particular advantage in it 

Lastly, the following proposition may be given which Playfair added as 
VI. E. It appears in the Data of Euclid, Prop. 93, and may be thus 
enunciated. 

If the angle BAG of a triangle ABC be bisected by the straight line AD 
meeting (he tirde circumscribed about the triangle in D, and if BD be joined, 
then 

the sum of BA, AC /> to AD as BC is to BD, 

Join CD. Then, since AD bisects the angle BAC, the subtended arcs 
SD, DC, and therefore the chords BD, DC, are 
equal. 

(i) The result can now be easily deduced from 
Ptolemy's theorem. 

For the rectangle AD, BC is equal to the sum of 
the rectangles AB, DC and .4C, BD, i.e. {since 
BD, CD are equal) to the ret tangle contained by 
BA + ACmdBD. 

Therefore the sum of BA, AC is to AD as BC 
is to BD. [vi. 16] 

(2) Euclid proves it differently in Data, Prop. 93, 

Let AD meet BC in JB. 

Then, since A£ bisects the angle BAC, 

BA i& to AC as BE to £C, [vi. 3] 

or, alternately, 

AB is to BE as AC to C£. [v. r6] 

Therefore also 

BA + AC is to BC BS AC to CE. [v, la] 

Again, since the angles BAD, EA C are equal, and the angles ADB, ACE 
are also equal, [in. 2i] 

the triangles ABD, A EC are equiangular. 

Therefore ^C is to CE as AD to BD. [vi, 4] 




ail BOOK VI [vi. 16, 17 

Hence BA^ACisXaBCiBADloBD, [v. 11] 

and, alternately, 

BA + AC is to AD as BC is to BD. (v. 16] 

Euclid concludes that, if the circle ABC is pven in magnitude, and the 
chord BC cuts off a segment of it containing a given angle (so that, by Data 
Prop. 87, 2fCand also BD are given in magnitude), , ,, ,^^ 

the ratio of BA + AC to AD is given, 

and further that (since, by similar triangles, BD is to D£ as ^4 C is to C£, 
vhi\t BA + AC is to BCsi&ACis to C£), 

the rectangle (BA + AC), D£, being equal to the rectangle BC, BD, is 
also given. 

Proposition 17 

// three straight lines be proportional, the rectangle con- 
tained by the extremes is equal to the square on the mean; 
and, if the rectangle contained by the extremes be equal to the 
square on the mean, the three straight lines will be proportional. 

Let the three straight lines A, B, C be proportional, so 
that, as A is to B, so is £ to C ; 

I say that the rectangle contained by A, C is equal to the 
square on B. 



Let D be made equal to B. 

Then, since, as A is to B, so is ^ to C, 

and B is equal to D, 

therefore, as A is to B, so is Z* to C, 

But, if four straight lines be proportional, the rectangle 
contained by the extremes is equal to the rectangle contained 
by the means. [vi, i6' 

Therefore the rectangle A, C h equal to the rectangle 
B,D. 

But the rectangle S, D is the square on B, for B is 
equal to D ; 

therefore the rectangle contained by A, C is equal to the 
square on B. 

Next, let the rectangle A, Che equal to me square on B 
I say that, as ^ is to B, so is B to C. 



VI. 17, i8] PROPOSITIONS 16—18 aaff 

For, with the same construction, 
since the rectangle A, C is equal to the square on B, 
while the square on B is the rectangle B, D, for B is equal 
to A 
therefore the rectangle A, C '\s equal to the rectangle B, D. 

But, if the rectangle contained by the extremes be equal 
to that contained by the means, the four straight lines are 
proportional. [vi. 16] 

Therefore, as A is to B, so is D to C 

But 5 is equal to /?; '•'' 

therefore, as A is to B, so is B to C. 

Therefore etc. q. e. d. 

VI. 17 is, of course, a particular case of vi. 16. 

Proposition 18. 

On a given straight line to describe a rectilineal figure 
similar and similarly situated to a given rectilineal figure. 

Let AB be the given straight line and CE the given 
rectilineal figure ; 

thus it is required to describe on the straight line AB a 
rectilineal figure similar and similarly situated to the recti- 
lineal figure CE. 





Let DF be joined, and on the straight line AB, and at 
the points A, B on it, let the angle GAB be constructed 
equal to the angle at C, and the angle ABG equal to the 
angle CDF. [1. 13] 

Therefore the remaining angle CFD is equal to the angle 
AGB ; [I. 32] 

therefore the triangle FCD is equiangular with the triangle 
GAB. 

Therefore, proportionally, as FD is to GB, so is FC to 
GA, and CD to AB. 



«39 BOOK VI [VL 18 

Again, on the straight line BG, and at the points B, G on 
it, let the angle BGH be constructed equal to the angle DFE, 
and the angle GBH equal to the angle FDE. [i. »3] 

Therefore the remaining angle at E is equal to the re- 
maining angle at H ; [i. 3^] 

therefore the triangle FDE is equiangular with the triangle 
GBH\ 

therefore, proportionally, as FD is to GB, so is FE to 
GH, and ED to HB. [vi. 4] 

But it was also proved that, as FD is to GB^ so is FC to 
GA, and CD to AB ; 

therefore also, as FC is to AG, so is CD to AB, and /^£' 
to G//, and further £"/? to HB. 

And, since the angle CFD is equal to the angle AGB, 

and the angle DFE to the angle ^G^//, 

therefore the whole angle CFE is equal to the whole angle 
AGH. 

For the same reason 

the angle CDE is also equal to the angle ABH. 
And the angle at C is also equal to the angle at A, 

and the angle at E to the angle at H. 
Therefore AH is equiangular with CE ; 
and they have the sides about their equal angles proportional ; 

therefore the rectilineal figure AH is similar to the 

rectilineal figure CE. [vi. Def. i] 

Therefore on the given straight line AB the rectilineal 
figure AH has been described similar and similarly situated 
to the given rectilineal figure CE. 

.. Q. E. F. 

Simson thinks the proof of this proposition has been vitiated, his grounds 
for this view being (1) that it is demonstrated only with reference to 
quadrilaterals, and does not show how tt may be extended to figures of five or 
more sides, (i) that Euclid infers, from the fact of two triangles being 
equiangular, that a side of the one is to the corresponding side of the other as 
another side of the first is to the side corresponding to it in the other, i.e. he 
permutes, without mentioning the fact that he does so, the proportions 
obtained in vi. 4, whereas the proof of the very next proposition gives, in a 
similar case, the intermediate step of permutation. I think this is hyper- 
criticism. As regards (a) it should be noted that the permuted form of the 
proportion is arrived at first in the proof of vi, 4 ; and the omission of the 



VI. 1 8] 



PROPOSITION i8 



'40- 



intermediate step of allertiandOy whether accidental or not, is of no importance. 
On the other hand, the use of this form of the proportion certainly simplifies 
the proof of the proposition, since it makes unnecessary the subsequent 
IX aequali steps of Simson's proof, their place being taken by the inference 
[v. 1 1] that ratios which are the same with a third ratio are the same with one 
another. 

Nor is the first objection of any importance. We have only to take as the 




given polygon a polygon of five sides at least, as CDEFG, pin one extremity 
of CD, say D, to each of the angular points other than C and E, and then 
use the same mode of construction as Euclid's for any number of successive 
triangles as ABL, LBK, etc, that may have to be made. Euclid's con- 
struction and proof for a quadrilateral are quite sufficient to show how to deal 
with the case of a figure of five or any greater number of sides. 

Clavius has a construction which, given the power of moving a figure 




bodily from one position to any other, is easier. CDEFG being the given 
polygon, join CE, CF. Place AB on CD so that A falls on C, and let B 
fall on ly, which may either lie on CD or on CD produced. 

Now draw DE parallel to DE, meeting CE, produced if necessary, in E, 
EF' parallel to EF, meeting CF, produced if necessary, in E", and so on. 

Let the parallel to the last side but one, FG^ meet CG, produced if 
necessary, in C. 

Then CUE FG is similar and similarly situated to CDEFG, and it is 
constructed on CD, a straight Hne equal to AB. 

The proof of this is obvious. 

A more general construction is indicated in the subjoined figure. If 
CDEFG be the given polygon, suppose its angular points all joined to any 
point O and the connecting straight lines produced both ways. Then, if CD, 
a straight line equal io AB, be placed so that it is parallel to CD, and C, D 
lie respectively on OC, OD (this can of course be done by finding fourth 
proportionals), we have only to draw UE, EF, etc., parallel to the 
corresponding sides of the original polygon in the manner shown. 



a$a 



BOOK VI 



[VL i8, 19 



De Morgan would rearrange Props. 18 and 20 in the following manner. 
He would combine Prop. 18 and the first part of Prop, ao into one, with the 
enunciation: 



E 




\FJ.S^-.:ifi-'*^^-.'.-~-----^ 




Pairs of similar triangits, similarly put tcgeihtr, give similar figures ; and 
every pair of similar figures is composed of pairs of similar triangles similarly 
put together. 

He would then make ihs problem of vi. 18 an application of the first part 
In form this would certainly appear to be an improvement; but, provided that 
the relation of the propositiotis is understood, the matter of form is perhaps 
not of great importance. 



Proposition 19. 

Similar triangles are to one another in the duplicate ratio 
of ike corresponding sides. 

Let ABC, DEFhe. similar triangles having the angle at 

B equal to the angle at £, and such that, as AB is to BC, so 

s is DE to EF, so that BC corresponds to EF; [v, Def. n] 

I say that the triangle ABC has to the triangle DEF a ratio 
duplicate of that which BC has to EF. 





For let a third proportional BG be taken to BC, EF, so 
that, as BC is to EF, so is EF to BG ; [vi. 1 1] 

10 and let AG be joined. 

Since then, as AB is to BC, so is DE to EF, 

therefore, alternately, as AB is to DE, so is BC to EF. [v. 16] 



VI, 19] PROPOSITIONS t8, 19 S33 

But, as BC is to EF, so is EF to BG ; ■" 
therefore also, as AB is to DE, so is EF to BG. [v. n] 

IS Therefore in the triangles ABG, DBF the sides about 
the equal angles are reciprocally proportional. 

But those triangles which have one angle equal to one 
angle, and in which the sides about the equal angles are 
reciprocally proportional, are equal; [vi. 15] 

30 therefore the triangle ABG is equal to the triangle DEF. 
Now since, as BC is to EF, so is EF to BG, 

and, if three straight lines be proportional, the first has to 
the third a ratio duplicate of that which it has to the second, 

[v, Def. 9] 
therefore BC has to BG a ratio duplicate of that which CB 
as has to EF. 

But, as CB is to BG, so is the triangle ABC to the 
triangle ABG ; [vi. i) 

therefore the triangle ABC also has to the triangle ABG a 
ratio duplicate of that which BC has to EF. 

30 But the triangle ABG is equal to the triangle DEF; 

therefore the triangle ABC also has to the triangle DEF a 
ratio duplicate of that which BC has to EF. ...... ,. ,. , 

Therefore etc. 

•I PoRiSM. From this it is manifest that, if three straight 
35 lines be proportional, then, as the first is to the third, so is 
the figure described on the first to that which is similar and 
similarly described on the second, 

Q. E. D. 

4. uid such that, as AB is \Q BC, so Is DE to BP, Utcrally "(tiiangls) having 
the anjfle at B equal to the anj;1e at f , and (AavtHg). as AB to BC^ so DK to EF^ 

Having combined Prop. [8 and the first part of Prop. 20 as just indicated, 
De Morgan would tack on to Prop. 19 the second part of Prop. 20, which 
asserts that, if similar polygons be divided into the same number of similar 
triangles, the triangles are " hemokgoui to the wholes " (in the sense that the 
polygons have the same ratio as the corresponding triangles have), and that 
the polygons are to one another in the duplicate ratio of corresponding sides. 
This again, though no doubt an improvement of form, would necessitate the 
drawing over again of the figure of the altered Proposition 18 and a certain 
amount of repetition. 

Agreeably to his su^estion that Prop, 23 should come before Prop. 14 
which is a particular case of it, De Morgan would prove Prop. 19 for 
parctUtlsgrams by means of Prop, 13, and thence infer the truth of it for 



'*^ BOOK VI [vi. 19 

triangles or the halves of the parallelograms. He adds ; " The method of 
Euclid is an elegant application of the operation requisite to compound equal 
ratios, by Which the conception of the process is lost sight of." For the 
general reason given in the note on vi, 14 above, I think that Euclid showed 
the sounder discretion in the arrangement which he adopted. Moreover it is 
not easy to see how performing the actual operation of compoutiding two 
equal ratios can obscure the process, or the fact that two equal ratios are 
being comfXJunded. On the definition of compounded ratios and duplicate 
ratio, De Morgan has himself acutely pointed out that "composition" is here 
used for the process of detecting the single alteration which produces the 
effect of two or more, the duplicate ratio being the result of compounding two 
equal ratios. The proof of vi. 19 does in fact exhibit the single alteration 
which produces the effect of two. And the operation was of the essence of 
the Cireek geometry, because it was the manipulation of ratios in this manner, 
by simplification and transformation, that gave it so much power, as every one 
knows who has read, say, Archimedes or ApoUonius. Hence the introduction 
of the necessary operation, as well as the theoretical proof, in this proposition 
seems to me to have been distinctly worth while, and, as it is somewhat 
simpler in this case than in the more general case of vj. 23, it was in 
accordance with the plan of enabling the difficulties of Book vi. to be more 
easily and gradually surmounted to give the simpler case first. 

That Euclid wished to emphasise the importance of the method adopted, 
as well as of the result obtained, in vi. 19 seems to me clearly indicated by 
the Porism which follows the proposition. It is as if he should say : "I have 
shown you that similar triangles are to one another in the duplicate ratio of 
corresponding sides; but I have also shown you incidentally how it is possible 
to work conveniently with duplicate ratios, viz. by transforming them into 
simple ratios between straight lines. I shall have occasion to illustrate the 
use of this method in the proof of vi. 22." 

Tlie Porism to VI. 19 presents one difficulty. It will be observed that it 
speaks of ^^ figure ((Bos) described on the first straight line and of that which 
is similar and similarly described on the second. If " figure " could be 
regarded as loosely used for the figure of the proposition, i.e. for a triangle, 
there would be no difficuity. If on the other hand " the figure " means any 
rectilineal figure, i.e. any p>olygon, the Porism is not really established until 
the next proposition, vi, 20, has been proved, and therefore it is out of place 
here. Yet the correction Tpiynivoi', triangle, for Ahot, figure, is due to Theon 
alone ; P and Campanus have " figure," and the reading of Philoponus and 
Psellus, TtTpayiiivov, square, partly supports (TS<n, since it can be reconciled with 
t'Sot but not with Tpiyuvoi'. Again the second Porism to vj. jo, in which this 
Porism is reasserted for any rectilineal figure^ and which is omitted by 
Campanus and only given by P in the margin, was probably interpolated by 
Theon. Heiberg concludes that Euclid wrote "figure" {<ISot), and Theon, 
seeing the difficulty, changed the word into " triangle " here and added For. 2 
to VI. «o in order to make the matter clear. If one may hazard a guess as to 
how Euclid made the slip, may it be that he first put it after vi 20 and then, 
observing that the expression of the duplicate ratio by a single ratio between 
two straight lines does not come in vi. so but in vi. rg, moved the Porism to 
the end of vi. 19 in order to make the connexion clearer, without noticing 
that, if this were done, tTSo! would need correction ? 

The following explanation at the end of the Porism is bracketed by 
Heiberg, viz. "Since it was proved that, as CB is to BG, so is the triangle 



vt. 19, 20] 



PROPOSITIONS 19, 20 



23s 



ABCio the triangle ABG, that is DEF." Such explanations in Porisms are 
not in Euclid's manner, and the words are tiot in Campanus, though they date 
from a time earlier than Theon. 



Proposition 20. 

Similar polygons are divided into similar triangles, and 

into triangles equal in multitude and in the same ratio as 

the wholes, and the polygon has to the polygon a ratio duplicate 

of that which the corresponding side has to the corresponding 

5 side. 

Let ABCDE, FGHKL be similar polygons, and let AB 
correspond to FG ; 

I say that the polygons ABCDE, FGHKL are divided into 
similar triangles, and into triangles equal in multitude and in 
10 the same ratio as the wholes, and the polygon ABCDE has 
to the polygon FGHKL a ratio duplicate of that which AB 
has to FG. 



Let BE, EC, GL, Z^ be joined. 



f.h 




Now, since the polygon ABCDE is similar to the polygon 
,5 FGHKL, 

the angle BAE is equal to the angle GFL ; 

and, as BA is to AE, so is GF to FL, [vi. Def. i] 

Since then ABE, FGL are two triangles having one 
angle equal to one angle and the sides about the equal angles 
K proportional, 

therefore the triangle ABE is equiangular with the triangle 

FGL ; [v.. 6] 

so that it is also similar ; [vi. 4 and Def. i] 

therefore the angle ABE is equal to the angle FGL. 



136 ■■•- BOOK VI :. [vt. 20 

»S But the whole angle ABC is also equal to the whole angle 
FGH because of the similarity of the polygons ; 

therefore the remaining angle EBC is equal to the angle 
L,GH. 

, And, since, because of the similarity of the triangles ABE, 
3PFGL, 

as EB is to BA, so is LG to GF, 
and moreover also, because of the similarity of the polygons, 

as AB is to BC, so is FG to GH, 
therefore, ex aequali, as EB is to BC, so is Z G^ to GH ; [v. aa] 

35 that is, the sides about the equal angles EBC, LGH are 
proportional ; 

therefore the triangle EBC is equiangular with the triangle 
LGH, [vi. 6] 

so that the triangle EBC is also similar to the triangle 
¥>LGH. [vi. 4 and Def. i] 

For the same reason 
the triangle ECD is also similar to the triangle LHK. 

Therefore the similar polygons ABCDE, FGHKL have 
been divided into similar triangles, and into triangles equal in 
45 multitude. 

I say that they are also in the same ratio as the wholes, 
that is, in such manner that the triangles are proportional, 
and ABE, EBC, ECD are antecedents, while FGL, LGH, 
LHK are their consequents, and that the polygon ABCDE 
so has to the polygon FGHKL a ratio duplicate of that which 
the corresponding side has to the corresponding side, that is 
AB to FG. 

For let AC, FH be joined. 

Then since, because of the similarity of the polygons, 

iS the angle ABC is equal to the angle FGH, 
and, as AB is to BC, so is FG to GH, 

the triangle ABC is equiangular with the triangle FGH ; 

[VI. 6] 

therefore the angle BAC is equal to the angle GFH, 
and the angle BCA to the angle GHF. 
60 And, since the angle BAM is equal to the angle GFN, 
and the angle ABM is also equal to the angle FGN, 



VI. jo] proposition lo 83J 

therefore the remaining angle A MB is also equal to the 
remaining angle FNG ; [i. 3a] 

therefore the triangle ABM is equiangular with the triangle 
6s FGN. 

Similarly we can prove that 
the triangle BMC is also equiangular with the triangle GNH, 
Therefore, proportionally, as AM is to MB, so is FN to 
NG, 
70 and, as BM is to MC, so is GN to NH ; 

so that, in addition, ex aequali, 

as AM is to MC, so is FN to NH. 

But, as AM is to MC, so is the triangle ABM to MBC, 

and AME to EMC; for they are to one another as their 

7S bases. [vi. i] 

Therefore also, as one of the antecedents is to one of the 

consequents, so are all the antecedents to all the consequents ; 

[v. is] 

therefore, as the triangle A MB is to BMC, so is ABE to 

CBE. 
80 But, as AMB is to BMC, so is AM to MC ; 

therefore also, as AM is to MC, so is the triangle ABE to 

the triangle EBC 

For the same reason also, 

as FN is to NH, so is the triangle FGL to the triangle 
8s GLH. 

And, as AM is to MC, so is FN to NH; 

therefore also, as the triangle ABE is to the triangle BEC, 

so is the triangle FGL to the triangle GLH ; 

and, alternately, as the triangle ABE is to the triangle FGL, 
90 so is the triangle BEC to the triangle GLH. 

Similarly we can prove, if BD, GK be joined, that, as the 

triangle BEC is to the triangle LGH, so also is the triangle 

ECD to the triangle LHK. 

And since, as the triangle ABE is to the triangle FGL, 
9S so is EBC to LGH, and further ECD to LHK, 

therefore also, as one of the antecedents is to one of the 

consequents, so are all the antecedents to all the consequents ; 

[v. 13 

therefore, as the triangle ABE is to the triangle FGL, 
so is the polygon ABCDE to the polygon FGHKL. 



i3S BOOK VI [vi. 10 

too But the triangle ABE has to the triangle FGL a ratio 

duplicate of that which the corresponding side AB has to the 

corresponding side FG\ for similar triangles are In the 

duplicate ratio of the corresponding sides. [vi. 19] 

Therefore the polygon ABCDE also has to the polygon 

■OS FGHKL a ratio dupUcate of that which the corresponding 

side AB has to the corresponding side FG. 

Therefore etc. 

PoRiSM, Similarly also it can be proved in the case of 
quadrilaterals that they are in the duplicate ratio of the 
110 corresponding sides. And it was also proved in the case of 
triangles; therefore also, generally, similar rectilineal figures 
are to one another in the duplicate ratio of the corresponding 
sides. 



Q. E, D. 



J. in the same ratio as the wholes. The Mune word J/iiXnTmi is used which 1 have 
generally translated by ^* corresponding.'^ But here it is followed by a dative. 5;4£X(^a r«r 
fiXeif *' Acmoio^Mi viih the wholes ," instead of being used ab^iotutely. The meaning can 
therefore here be nothing else but 'Mn the same ratio with" or "proportional to the 
wholes"^ and Euclid seems to recognise that he is making a special use of the word, 
because he eitplains it lower down (1. 46) : "the triangles are homolc^ous to the wholes, thai 
is, in such manner that the triangles are proportional, and AB£j EBC, ECD are ante- 
cedents, while FGL, LGH, I. UK are their consequents," 

49. iv6fu¥a rr0Twf, " iA^tr consequents," is a little awkward, but may be supposed to 
indicate which triangles curtespond to which ai consctjuent to antecedent. . ,., , . 

An alternative proof of the second part of this proposition given after the 
Porisms is relegatetj by August and Heiberg to an Appendix as an interpolation. 
It is shorter than the proof in the text, anti is the only one given by many 
editors, including Clavius, Billingsley, Barrow and Simson. It runs as follows: 

" We will now also prove that the triangles are homolc^ous in another and 
an easier manner. 





Again, let the polygons ABCDE, FGHKL be set out, and let BE, EC, 
GL, Z//^ be joined 

I say that, as the triangle A BE is to FGL, so is EBC to LGH and ODE 
to HKL. 

For, since the triangle ABE is similar to the triangle FGL, the triangle 
ABE has to. the triangle FGL a ratio duplicate of that which BE has to GL, 



VI, 30, ai] PROPOSITIONS ao, it *$g. 

For the same reason also 
the trkngte BEC has to the triangle GLIf a ratio duplicate of that i^ich 
£E has to GL. 

Therefore, as the triangle ABE is to the triangle EG£, so is BEC 
10 GZff. 

Again, since the triangle EBC \s similar to the triangle LGH, 

EBC has to LGH a ratio duplicate of that which the straight line CE has 
to HL. 

For the same reason aiso 

the triangle ECD has to the triangle LHK a ratio duplicate of that which 
CE has to HL. 

Therefore, as the triangle EBC is to LGH, so is ECD to LHK. 
But it was proved that, 

as EBC is to LGH, so also is ABE to FGL. 

Therefore also, as ABE is to FGL, so is BEC to GLH and ECD to 
LHK. • . 

•• ■ ' It 

Q. E. D. 

Now Euclid cannot fail to have noticed that the second part of his 
proposition could be proved in this way. It seems therefore that, in giving 
the other and longer method, he deliberately wished to avoid using the result 
of VI. 19, preferring to prove the first two parts of the theorem, as they can be 
proved, independently o( any relation between the areas of similar triangles. 

The first part of the Porism, stating that the theorem is true oi quadriiaitrals, 
would be superfluous but for the fact that technically, according to Book i. 
Def, 19, the term "polygon "{or figure of many sides, iroXuTrXtupoi') used in the 
enunciation of the proposition is confined to rectilineal figures of more than 
four sides, so that a quadrilateral might seem to be excluded. The mention 
of the triangle in addition fills up the tale of " similar rectilineal figures." 

The second Porism, Theon's interpolation, given in the text by the editors, 
but bracketed by Heiberg, is as follows : 

"And, if we talce O a third proportional to AB, EG, then BA has to O a 
rtUio duplicate of that which AB has to EC. 

But the polygon has also to the polygon, or the quadrilateral to the 
quadrilateral, a ratio duplicate of that which the corresponding side has to 
the corresponding side, that is AB to EG; 
and this was proved in the case of triangles aiso ; 

so that it is also manifest generally that, if three straight lines be proportional, 
as the first is to the third, so will the figure described on the first be to the 
similar and similarly described figure on the second." 



Proposition 21. • . ■ • 

Figures which are similar to the same rectilineal figure 
are also similar to one another. 

For let each of the rectilineal figures A,B]x. similar to C\ 
1 say that A is also similar to ^. 



«4o BOOK VI [vi. 31, It 

For, since A is similar to C * 

it is equiangular with it and has the sides about the equal 

angles proportional. [vi. Def. i] 



tn 





Again, since B is similar to C, 

it is equiangular with it and has the sides about the equal 
angles proportional, 

Therefore each of the figures A, B is equiangular with C 
and with C has the sides about the equal angles proportional; 

therefore A is similar to B, 

Q. E, D, 

It will be observed that the text above omits a. step which the editions 
generally have before the final inference " Therefore A is similar to B." The 
words omitted are "so that A is also equiangular with B and [with B] has the 
sides about the equal angles proportional." Heiberg follows P in leaving 
them out, conjecturing that they may be an addition of Th eon's. 



Proposition 22. 

J/ /our straight lines be proportional, the rectilineal figures 

similar and similarly described upon them -will also be pro- 
portional ; and, if the rectilineal figures similar and similarly 
described upon them be proportional, the straight lines will 
themselves also be proportional. 

Let the four straight lines AB, CD, EF, GH be pro- 
portional, 

so that, as ^.ff is to CD, so is EFxo GH, 
and let there be described on AB, CD the similar and similarly 
situated rectilineal figures KAB, LCD, 
and on EF, GH the similar and similarly situated rectilineal 
figures J/A, NH; 
I say that, as KAB is to LCD, so is MF to NH. 

For let there be taken a third proportional O to AB, CD, 
and a third proportional fi to EF, GH. [vi. n] 



VI. »a] PROPOSITIONS ai, »a J4|, 

Then since, as AB is to CD, so is EF to GH, 
and, as CZ? is to O, so is GH to /*, 
therefore. Mr aequali, as Wi? is to O, so is EF to P. [v. u] 

But, as -f4^ is to t>, so is AVJ^ to ZCZ>, 

and, as EF Is to /", so is MF to A^^; ^'''' "'' **''^ 

therefore also, as KAB is to Z,CA so is MF to A''//, [v. n] 





/7 



Next. let MF be to A^//^ as KAB is to ZLCZ? ; 
I say also that, as ^4 5 is to CD, so is EFto GH. 

For, if EF is not to GH as ^^ to CZ?, 

let j£"/^be to jjj? as AB to CA [vi. u] 

and on QR let the rectilineal figure SR be described similar 
and similarly situated to either of the two MF, NH. [vi. i8] 

Since then, as AB is to CD, so is EF to QR, 
and there have been described on AB, CD the similar and 
similarly situated figures KAB, LCD, '- 

and on EF, QR the similar and similarly situated figures 
MF, SR, 

therefore, as KAB is to LCD, so is MF to SR. 
But also, by hypothesis, 

as KAB is to L CD, so is MF to NH ; 
' ■ therefore also, as MF is to SR, so is MF to A''//^ [v. 1 1] 

Therefore MF has the same ratio to each of the figures 
NH,SR; 

therefore NH is equal to SR. [v. 9] 

But it is also similar and similarly situated to it ; 
therefore GH is equal to QR. 



S4» BOOK VI '< ' [vi. ** 

And, since, as AB is to CD, so is £F to QJ?, 

while QH is equal to G//, 

therefore, as AB is to CD, so is EFto GH. " '" 

Therefore etc. 

A , ' • / t . Q. E. D. 

The second assumption in the first step of the first part of the proof, vi^. 
that, as CD is to O, so GM to J", should perhaps be explained. It is a 
deduction [by v. ii] from the facts that 

-. AMis to CJ? as CD to O, 

£I!'is to Gffas GUto P, 

Mid •- AB is to CD as EFta GH. 

The defect in the proof of this proposition is well known, namely the 
assumption, without proof, that, because the figures NH, SR are equal, 
besides being similar and similarly situated, their corresponding sides GH, QR 
are equai. Hence the minimum addition necessary to make the proof 
complete is a proof of a lemma to the effect that, iffim similar figurn are also 
equal, any pair of corresponding sides are equal. 

To supply this lemma is one alternative ; another is to prove, as a 
preliminary proposition, a much more general theorem, viz. that, tf the 
duplicate ratios of two ratios an equal, the tnio ratios are themselves equal. 
When this is proved, the second part of vi, 12 is an immediate infeience from 
it, and the effect is, of course, to substitute a new proof instead of 
supplementing Euclid's. 

I. It is to be noticed that the lemma requitt:d as a mini mum is very like 
what is needed to supplement vi. 28 and 19, in the proofs of which Euclid 
assumes that, if tivo similar parallelograms are unequal, any side in the greater 
is greater than the corresponding side in the smaller. Therefore, on the whole, it 
seems preferable to adopt the alternative of proving the simpler lemma which 
will serve to supplement all three proofs, vii, that, if of two similar rectilineal 
figures the first is greater than, equal to, or less thaft, the second, any side of the 
first is greater than, equal to, or less than, the corresponding side of the second 
respectively. 

The case of equality of the figures is the case required for vi. t2 ; and the 
proof of it is given in the Greek text after the proposition, Since to give such 
a " lemma " after the proposition in which it is required is contrary to Euclid's 
manner, Heiberg concludes that it is an interpolation, though it is earlier than 
TheotL The lemma runs thus ; 

"But that, if rectilineal figures be equal and similar, their corresponding 
sides are equal to one another we will prove thus. 

Let Nff, SR be equal and similar rectilineal figures, and suppose that, 
as HG is to GN, so is RQ to QS', 
I say that RQ is equal to HG. 

For, if they are unequal, one of them is greater; 
let ^Q be greater than HG. 



VI. sa] PROPOSITION 31 _s4S 

Then, since, as ^0 is to QS, so is I/G to GNi 
alternately also, as ^Q is to //G, so is QS to GJVi 
and Q£ is greater than J/G ; 

therefore QS is also greater than GN; 

so that /IS is also greater than HN*. 

But it is also equal : which is impossible; ' " ' 

Therefore QR is not unequal to GH\ •. vk D (•' 

therefore it is equal to it," ^«^f.^ ,> 

[The step marked • is easy to see if it is remembered that it is only 
necessary to prove its truth in the case of triangles (since similar polygons are 
divisible into the same number of similar and similarly situated triangles 
having the same ratio to each other respectively as the polygons have). If the 
triangles be applied to each other so that the two corresponding sides of each, 
which are used in the question, and the angles included by them coincide, 
the truth of the inference is obvious.] 

The lemma might also be arrived at by proving that, ^« ratio is greater than 
a ratio of equality, tht ratio which «> its dupHeate is also greater than a ratio of 
equality ; and if ike ratio lohich is duplicate of anotfur ratio is greater than a 
ratio of equally, the ratio of which it is the duplicate is also greater than a ratio 
of equality. It is not difficult to prove this from the particular case of v. 25 in 
which the second magnitude is equal to the third, i.e. from the fact that in 
this case the sum of the extreme terms is greater than double the middle term. 

II. We now come to the alternative which substitutes a new proof for the 
second part of the proposition, making the whole proposition an immediate 
inference from one to which it is practically equivalent, viz. that 

(i) // duo ratios be equal, their duplicate ratios are equal, and (2) cott' 
versely, if the duplicate ratios of two ratios be equal, the ratios are equal 

The proof of part (i) is after the manner of Euclid's own proof of the first 
part of VI, a*. 

Let j1 be to .5 as C to A 
and let Jf be a third proportional to A, B, and Fa third proportional to C, D, 
so that 

A '\% \!0 B »& B to X, 

and C is to Z) as 2J to Y\ 

whence A'\%\a X in the duplicate ratio of A to B, 

and C is to Y in the duplicate ratio of C to D. 

Since j4 13 to i? as C is to A 

and ^ is to Jf as /i is to .5, 

i.e. as C is to Z*, , , 

IV. Ill 

i.e as /J is to K, . ^ ^ 

therefore, ex aequali, /f is to Jf as C is to K -• 

Part (2) is much more difficult and is the crux of the whole thing. 

Most of the proofs' depend on the assumption that, B being any magnitude 
and P and Q two magnitudes of the same kind, there does exist a magnitude 
A which is to S in the same ratio as /' to Q, It is this same assumption 



«44 '■'■ BOOK VI [vi. ai 

which makes Euclid's proof of v. i8 illegitimate, since it is nowhere proved 
in Book v. Hence any proof of the proptosition now in question which 
involves this assumption even in the case where B, P, Q are all straight lines 
should not properly be given as an addition to Book v. ; it should at least be 
postponed until we have leamt, by means of vi. iz, giving the actual 
construction of a fourth proportional, that such a fourth proportional exists. 

Two proofs which are given of the proposition depend upon the following 
lemma. 

^A, B, C ^ tAru magnitudes of ofte kind, and D, E, F three Mitgnitudes 
of one kind, then, if 

Ike ratio of A fa B is greater than that of D toE, i ' 

and tht ratio of "^ to Q, greater than that of 'R to'F, ' 

ex aequali, the ratio of A to C is greater than thai of V to F. 

One proof of this does not depend upon (he assumption referred to, and 
therefore, if this proof is used, the theorem can be added to Book v. The 
proof is that of Hauber (Camerer's Euclid, p. -358 of Vol. u.) and is reproduced 
by Mr H. M. Taylor. For brevity we will use symbols. 

Take equimultiples m^, mD of A, D and nB, nE of B, E such that 

mA>nB, but mD'^nE. 

Also let pB, pE be equimultiples of B. E and qC, qP equimultiples of 
C, .f such that 

pB>qC, hai pElf-gF. 

Therefore:, multiplying the first tine by^ and the second by n, we have 
pmA>pnB,pmD1(-pnE, i. 

and npS>nqC,npE1f-nqF, 

whence pmA>nqC, pmDI^nqF. ' \ 

Now P*nA, pmD are equimultiples of mA, mU, 

and nqC, ffl?/^ equimultiples of qC, qF. 

Therefore [v. 3] they are respectively equimultiples of A, D and of C, F. 

Hence [v. Def. 7] A : C>D : F. 

Another proof given by Claviua, though depending on the assumption 
referred to, is neat 
Take G such that 

GiC = E'.F. • ■- 



A 




B 
A 























Therefore 
and 

Therefore 


«i . 


■.C>G.C, 
B>G. 
■.G>AiB. 



[v. 13] 
[v. 10] 
[v. 8] 



VI. ai] PROPOSITION 22 145 



But 




A ; 


3>DiE. 


Therefore, a fortiori. 


A : 


G>D:E. 


Suppose ^ taken such that 


H: 


G= D:B. 


Therefore 






A>H. 


Hence 




A : 


: C>H'. C. 


But 




H. 


.G = D:£, 






G . 


C^E:F. 


Therefore, ex 


aequali^ 


J5f 


; C=£}.K 


Hence 




A : 


\C>D.R 



[V. 


'3. 


.0] 




[V 


.8) 




[V. 


2.] 




[V- 


■3] 



Now we can prove that 
Ratios of whUk equai ratios are duplitate are equal. 



Suppose that 


A : 


.B = B: 


C, 


and 


D 


:£ = £ 


:iS 


and further that 


A : 


; €=£>: 


E 


it is required '0 prove that 









A B^D-.E. 

For, if not, one of the ratios must be greater than the othur. 
I>et A : B\x^t greater. 

Then, since A : B = B : C^ 

and £>:E = E:E, 

while A: B>D: E, 

it follows that B: C>E\F. [v- 13] 

Hence, by the lemma, ex aequali, 

A : C>D -.F, 

which contradicts the hypothesis. 

Thus the ratios A : B and D : E cannot be unequal j that is, they are equal. 

Another proof, given by Dr Lachlan, also assumes the existence of a 
fourth proportional, but dei:)ends upon a simpler lemma to the effect that 

// is impussible that two different ratios can have the same duplicate ratio. 

For, if possible, let the ratio ^ : ^ be duplicate both ol A -.X and A : V, 
so that 



[V.8] 

[v. II, 13] 

[V. lo] 









A 


■.X = 


X 


■B, 


and 






A 


: y= 


: V 


■B. 


Let X be greater 


than 


y. 










Then 






A: 


x< 


A : 


y-. 


that is, 






X . 


B< 


Y: 


-e, 


or 








x< 


Y. 




But X is greater 


than 


Y: 


which IS a 


Lbsurd, etc. 


Hence 








x= 


y. 





%t6 


BOOK VI 


Now suppose that 


A : B^B: C, 




JDiE = E .F, 


and 


A : C = D:F. 


To prove that 


A\B = D:E. 


If this is not so, suppose 


that 




A:B = D:Z. 


Since 


A.C^D.F, 


therefore, inversely, 


C:A=F:D. 


Therefore, «c aequali. 






C: JS = F:2, 


or, inversely. 


B: C = Z: F. 


Therefore 


A.B = Z:F. 


But 


A : B ^ D : Zy^yj hypothesis. 


Therefore 


D:Z=Z. F. 


Also, by hypothesis, 


D:E=E:F; 


whence, by the lemma, 


E = Z. 


Therefore 


A-.B-^D-.E. 



[VI. 



[v. J a] 
[v. m] 
[V. ..J 



De Morgan remarks that the best way of remedying the defect in Euclid 
is to insert the proposition {the lemma to the last proof) that // is imposiihU 
thai two differml ratios can have the same duflitate ratio, "which," he says, 
"immediately proves the second (or defective) case of the theorem." But this 
seems to be either too much or too little : too much, if we choose to make 
Che minimum addition to Euclid (for that addition is a lemma which shall prove 
that, if a duplicate ratio is a ratio of equality, the ratio of which it is duplicate 
is also one of equality), and too little if the proof is to be altered in the more 
fundamental manner explained above, 

I think that, if Euclid's attention had been drawn to the defect in his 
proof of VI. 2 2 and he had been asked to remedy it, he would have done so 
by supplying what I have called the minimum lemma and not by making the 
more fundamental alteration. This I infer from Prop, 24 of the Data, where 
he gives a theorem corresponding to the proposition that ratios of which equal 
ratios, are duplicate are equal. The proposition in the Data is enunciated 
thus : If three straight tines be proportional, and the first have to the third a 
given ratio, it wilt also have to the second a given ratio. 

A, B, C being the three straight lines, so that 

A .B = B.C, 

and A : C being a given ratio, it is required to prove that A : B vi also a 

given ratio. 

Euclid takes any straight line D, and first finds another, F, such that 
D .F=A -.C, 
whence D : F must be a given ratio, and, as ^ is given, F is therefore given. 

Then he takes E a mean proportional between D, P, so that 

D .£ = E:F. 



VI. J J, 23] PROPOSITIONS 2t, 13 f||> 

It follows [vi. 17] that 
the rectangle D, Ph equal to the square on E. 
But D, Fzxe both given ; 

therefore the square on E is given, so that E is also given. 
[Observe that De Morgan's lemma is here assumed without proof. It 
may be proved {ij as it is by I>e Morgan, whose proof is that given above, 
p. 345, (i) in the manner of the "minimum Semma," pp. 342 — 3 above:, or 
(3) as it is by Proclus on i. 46 (see note on that proposition).] 
Hence the ratio D \ E\s given. 
Now, since A : C= D : J<, 

and A : C= (square an A): (rect. A, C), 

while D : F~ (square on D') : (tect. D, F), [vi. i] 

therefore (square on A) ; (rect. A, C)~ {square on D) • (rect. D, F). [v. 11] 
But, since A -. B = B : C, (rect. A, C) - (sq. an S); [vi. 17] 

and (rect. £>, F) = (sq. on E}, from above ; 

therefore (square on A) : (square on ^) = (sq, on i?) : {sq. on £). ' ■ ' 

Therefore^ says Euclid, • 1 w 

A;B = D:E, 
that is, ht assumes the truth of •i\. a fat squarts. 

Thus he deduces his proposition from vi, 21, instead of proving vi. 33 by 
means of it (or the corresponding proposition used by Mr Taylor and 
Or I^chlan). 

■i/u . .' .• > .:,.•-. Proposition 23. '■' ' "' 

Equiangular parallelograms have to one another the ratio 
compounded of the ratios of their sides. 

Let AC, CF be equiangular parallelograms having the 
angle BCD equal to the angle ECG ; 
jl say that the parallelogram AC has to the parallelogram 
CF the ratio (;ompounded of the ratios of the sides. 

7 

K B' 

U 

M 



For let them be placed so that BC is in a straight line 
with CG ; 

therefore DC is also in a straight line with CE. 
I Let the parallelngram DG be completed ; 
let a straight line K be set out, and let it be contrived that, 

as BC is to CG, so is i^ to Z. , 
and, as DC is to CE, so is L to M. [vi. 12] 



348 BOOK VI [vi. »3 

Then the ratios of K to L and oi L, Xo M are the same 
ij as the ratios of the sides, name]y of BC to CG and of DC 
to CE. 

But the ratio of A' to ^ is compounded of the ratio of K 
to L an<i of that of Z to M; 

so that JC has also to M the ratio compounded of the ratios 
ao of the sides. 

Now since, as BC is to CG, so is the parallelogram AC 
to the parallelogram C//, [vi. i] 

while, as BC is to CG, so is A" to Z, 

therefore also, as K is to L, so is AC to C//. [v. n] 

as Again, since, as Z?C is to C£, so is the parallelogram C// 
to CjS [vi. r] 

while, as DC is to C£, so is L to ^, 

therefore also, as Z. is to M, so is the parallelogram C/f to 
the parallelogram CK [v, 1 1] 

30 Since then it was proved that, as /f is to Z, so is the 
parallelogram AC to the parallelogram C//, 

and, as Z is to ^, so is the parallelogram CH to the 
parallelogram CB", 

therefore, mt ae^uali, as A' is to M, so is AC to the paralle lo- 
ss gram CF. 

But A" has to M the ratio compounded of the ratios of 

the sides ; 

therefore A C also has to CF the ratio compounded of the 

ratios of the sides. 

40 Therefore etc, 

Q. E. D. 

1,6, 19, 36. the ratio compounded of the ratios of the sides, M^w riy rirrmifunr 
4t T«>' TrXtvpi^ which, nieining literally " the ratio coropounded 0/ (it lijes," is negligently 
writlen here and commonly for Xiryav rir rvyKtif^trov in t«v tup i[\xvpi4» (sc. X^yuv}* 

II. let it be contrived that, as BC is to CO, so is K to L, The Greek phraiie in 
of the usual ter^e kind, untranslatable literally : xol yeyofirw ajt ftir ^ EF Tpit rip TE, 
offrwi 17 K irpij rf A, the words meaning "and let (there) be made, u BC to CG, sa Kio 
Z," where L is the straight line which has to be constructed. 

The second tJefinirion of the Daia says that A ratio is said to fe given if 
we can find (Tropi'o-ao-flot) [another ratio that />] the same with it. Accordingly 
VI, 13 not only proves that equiangular parallelograms have to one another a 
ratio which is compounded of two others, but shows that that ratio is "given" 
when its component ratios are given, or that it can be represented as a simple 
ratio between straight lines. 



VI. i3] PROPOSITION »3 94^ 

Just as vt. 13 exhibits the o[>eration necessary for eomfounding two 
ifttios, a proposition (8) of the Data indicates the operation by which we may 
divide one ratio hy another. The proposition proves that Thingt which 
have a given raiio to the same thing have also a grven ratio to one another. 
Euclid's procedure is of course to comf>ound one ratio vrith the inverse of the 
other ; but, when this is once done and the result of Prop. 8 obtained, he 
uses the result in the later propositions as a substitute for the method of 
composition. Thus he uses the division of i^tios, instead of composition, 
in the propositions of the Data which deal with the same subject-matter as 
VL 13. The effect is to represent the ratio of two equiangular parallelc^rams 
as a ratio between straight lines one of which is one side of one of the 
paraiMograms, Prop. 56 of the Data shows us that, if we want to express 
the ratio of the parallelogram AC Xx> the paraDelogram CF in the figure 



/■ 



ov. ur.' 

'.I ( 

..I '■■ ' 

.... ■■ , A'' 

of VI. 23 in the form of a ratio in which, for exampie, the side flC is the 
antecedent term, the required i-atio of the parallelograms is BC ; X-, where 

DC:CE=CG: X, 
or A" is a fourth proportional to DCarA the two sides of the parallelogram CF. 
Measure CK along CB, produced if necessary, so that 
DC .CE=CG . CK 
(whence CK is equal to X). * 

[This may be simply done by joining DG and then drawing EK parallel 
to it meeting CB in /T.] 

Complete the parallelogram AK. 
Then, since DC : CE ^ CG : CK, 

the parallelc^rams DK, C^are equal. [vi. 14] 

Therefore (AC).{CF) = {AC).(DK) > ■ [v. 7] 

= BC .CK ' [VI. i] 

^BC.X. 

Prop. 68 of the Data uses the same construction to prove that. If two 
equiangular parallelograms havt to one another a given ratio, and one side have 
to one side a given ratio, the remaining side will also have to the remaining side 
a given ratio. 

I do not use the figure of the Data but, for convenience' sake, I adhere 
to the figure given above. Suppose that the ratio of the parallelograms is 
given, and also that of CD to CE. 

Apply to CD the parallel<^ram DK equal to CE and such that CK, CB 
coincide in direction. [i. 45] 

TTien the ratio of AC to KD is given, being equal to that of ACio CF. 
And {AC) : (KB) = CB : CK; 

therefore the ratio of CB to CK is given. 



>SO BOOK VI [vi. ^3 

But. since KD= CF, 

Cn-.CE^CG: CK. [vi. 14] 

Hence CG : CK'xs a given ratio. 

And CB : CK was proved to be a given ratio. 

Therefore the ratio of CB to CG is given. \Data, Prop. 8] 

lastly we may refer to Prop. 70 of the Data, thL- first part of which proves 
what corresponds exactly to vi. 33, namely that, // in ttvo fquiangular paral- 
lelograms Ihe sides containing tht equal angles ham a given ratio to one atmlher 
[i.e. one side in one to one side in the other], the paralUlograms themselves will 
also have agiivn rat is to one another. [Here the ratios of BC to CG and of 
CD to CE are given.] 

The construction ts the same as iti the last case, and we have KD equal 
to CF^ so that 

CD . CE=CG : CK. [vi. 14] 

But the ratio of CD to CE is given ; 
therefore the ratio of CG to C^is given. 

And, by hypothesis, the ratio of CG to CB is given. 

Tlierefore, by dividing the ratios [ZJd/ir, Prop. 8], we see that the ratio of 
CB to CK^ and therefore [vj, i] the ratio of .^C to DK, or of AC to CF, 
is given. 

Euchd extends these propositions to the case of two parallelograms which 
have givfn but not equal angles. 

Pappus (v[i. p. 928) exhibits the result of vi. 13 in a different way, 
which throws new light on compounded ratios. He proves, namely, that a 
paratUlograin is to an equiangnlar parallelogram as the rectangle amtained hy 
tlu adjacent sides of the first is to the rectangle contained by the adjacent sides 
oj the second, 

A 

o 



C E 

t-et A C, OF he equiangular parallelograms 01 r the bases BC, EF, and let 
the angles at .5, £ he equal. 

Draw perjjendiculars AG, DH \.o BC, £/" respectively. 
Since the angles at B, G are equal to those at E, H, 

the triangles ABG, DEHsliu equiangular. 
' Therefore BA : AG ED: DM. [vi. 4] 

But BA 'AG = (rect. BA, BC) : (rect. AG, BC), 

and ED : DH ^ (rect, ED, EF) : (rect DH, EF). [vi, i] 

Therefore [v. [ 1 and v. 16] 

(TticX.. AB, BC) : (rect. DE, EF) = (rect. AG, BC) : (rect DH, EF) 

= {AC).{DF). 

Thus it is proved that the ratio compounded of the ratios A3 : DE and 
BC : EF is equal to the ratio of the rectangle AB, BC to the rectangle 
DE, EF. 



VL 83, 34] PROPOSITIONS 23, 14 *St 

Since each parallelogram in the figure of the proposition can be divided 
into pairs of equal triangles, and all the triangles which are the halves of either 
parallelogram have two sides respectively equal and the angles included by 
them equal or supplementary, it can be at once deduced from vt. 23 (or it 
can be independently proved by the same method) that triangles which have 
ont angle of the one equal or supplementary to one angle of the other are in (he 
ratio (ompoa tided of the ratios of the sides about the eqval or supplementary 
angles. Cf- Pappus vii. pp. 894—6. 

VI. 23 also shows that rectangles, and therefore parallelograms or triangles, 
are to one another in the ratio compounded of the ratios of theit bases and 
heights. 

The converse of vi. 23 is also true, as is easily proved by reductio ad 
absurdum. More generally, if two parallelograms or triangles are in the ratio 
compounded of the ratios of two adjacent sides^ the angles included by those sides 
are either equal or supplementary. 

Proposition 24. 

In any parallelogram the parallelograms about the diameter 
are similar both to the whole and to one another. 

Let ABCD be a parallelogram, and AC its diameter, 
and let EG, HK be parallelograms 
about AC\ 

I say that each of the parallelograms 
EG, HK is similar both to the whole 
ABCD and to the other. 

For, since EF has been drawn 
parallel to BC, one of the sides of the 
triangle ABC, 

proportionally, as BE is to EA, so is CF to FA. [vi. 2] 

Again, since FG has been drawn parallel to CD, one of 
the sides of the triangle ACD, 

proportionally, as CF\^ to FA, so is DG to GA. [vi. 2], 

But it was proved that, 

as CF is to FA, so also is BE to EA ; 

therefore al.so, as BE is to EA, so is DG to GA, 
and therefore, componendo, 

as BA is to AE, so is DA to AG, \y. 18] 

and, alternately, 

as BA is to AD, so is EA to AG. [v. 16] 

Therefore in the parallelograms ABCD, EG, the sides 
about the common angle BAD are proportional. 

And, since GF is parallel to DC, 




»j^ , BOOK VI [vi. 24 

the angle AFG is equal to the angle DC A ; 
and the angle DAC is common to the two triangles ADC, 
AGF, 

therefore the triangle ADC is equiangular with the triangle 
AGF. 

For the same reason 

the triangle ACB is also equiangular with the triangle 
AFE. 

and the whole parallelogram ABCD is equiangular with the 
parallelogram EG. 

Therefore, proportionally, 

as AD is to DC, so is v4G^ to GF, 

as DC is to CA, so is GF to FA., 

as ^C is to CB, so is AF to FE, 

and further, as CB is to BA, so is FE to EA. 

And, since it was proved that, 
as DC is to CA, so is GF to /v?, 
and, as AC is to C^, so is AF to /^£, 
therefore, ex aequali, as Z?C is to CB, so is GF to 7^/5'. [v. j^J 

Therefore in the parallelograms ABCD, EG the sides 
about the equal angles are proportional ; 

therefore the parallelogram ABCD is similar to the parallelo- 
gram EG. [vi. Def. i] 

For the same reason 
the parallelogram ABCD is also similar to the parallelogram 
KH', 

therefore each of the paralleJograms EG, HK is simitar to 
ABCD. 

But figures simlliar to the same rectilineal figure are also 
similar to one another ; [vi. ai] 

therefore the parallelogram EG is also similar to the parallelo- 
gram HK. 

Therefore etc. 

Q, E, D. 

Simson was of opinion that this proof was made up by some unskilful 
editor out of two others, the first of which proved by parallels (vi. 2) that 
the sides about the common angle in the parallelograms are proportional, 
while the other used the similarity of triangles {vi. 4.). It is of course true 



VI. 24. 25] PROPOSITIONS 34, 35 »S3 

that, when we have proved by vi. 2 the fact that the sides about the common 
angle are proportional, we can infer the proportionality of the other sides 
directly from 1, 34 combined with v. 7. But it does not seem to me unnatural 
that Euclid should (i) deliberately refrain from making any use of t. 34 and 
(3) determine beforehand that he would prove the sides proportional in a 
difinite ordtr beginning with the sides EA, AG and BA, AD about the 
common angle and then taking the remaining sides in the order indicated 
by the order of the letters A, G, F, E. Given that Euclid started the proof 
with such a fixed intention in his mind, the course taken presents no difficulty, 
nor is the proof unsystematic or unduly drawn out. And its genuineness 
seems to me supported by the fact that the proof, when once the first two 
sides about the common angle have been disposed of, follows closely the 
order and method of vi. 18. Moreover, it could readily be adapted to the 
more general case of two p>olygons having a common angle and the other 
corresponding sides respectively parallel. 

The parallelograms in the proposition are of course similarly situated as 
well as similar; and those "about the diameter" may be "about" the 
diameter produced as well as about the diameter itself 

From the first part of the proof it follows that parallelograms which have 
one an^te equal to one angle and the sides about those angles proportional 
are similar. 

Prop. 26 is the converse of Prop. 24, and there seems to be no reason 
why they should be separated as they are in the text by the interposition of 
VI. %i. Campanus has vi. 24 and 26 as vi. 22 and 23 respectively, vi. 23 as 
VI, 14, and VI. 35 as we have it. 

Proposition 25. 

To construct one and the same figure similar to a given 
rectilineal figure and equal to another given rectilineal figure. 

Let ABC be the given rectilineal figure to which the 
figure to be constructed must be similar, and D that to which 
it must be equal ; 

thus it is required to construct one and the same figure similar 
to ABC and equal to D. 





Let there be applied to BC the parallelogram BE equal 
to the triangle ABC [1. 44], and to CE the parallelogram CM 
equal to D in the angle FCE which is equal to the angle 
CBL. [1. 45] 



»$♦ BOOK Vr [vi. 35 

• Therefore BC is in a straight line with CF, and LE with 
EM. 

Now let GH be taken a mean proportional to BC, CF 
[vL 13J, and on GHlel KGHh& described similar and similarly 
situated to ABC. \y\. 18] 

Then, since, as BC is to GH, so is GH to CF, 
and, if three straight lines be proportional, as the first is to 
the third, so is the figure on the first to the similar and 
similarly situated figure described on the second, [vi. 19, For,] 
therefore, as BC is to CF, so is the triangle ABC to the 
triangle KGH. 

But, as BC is to CF, so also is the parallelogram BE to 
the parallelogram EF. [vi. i] 

Therefore also, as the triangle ABC is to the triangle 
KGH, so is the parallelogram BE to the parallelogram EF \ 
therefore, alternately, as the triangle ABC is to the parallelo- 
gram BE, so is the triangle KGH to the parallelogram EF. 

[v. 16] 

But the triangle ABC is equal to the parallelogram BE ; 
therefore the triangle jf 6'//' is also equal to the parallelogram 
EF. 

But the parallelogram EF is equal to D \ 
therefore KGH is also equal to D, 

And KGH is also similar to ABC. 

Therefore one and the same figure KGH has been con- 
structed similar to the given rectilineal figure ABC and equal 
to the other given figure D. 

Q, E, D. 

3. to which the flgiire to be const ructed muBt be similar, literally " lo which it 
Is requiied to construct (one) Eimiisr," ^ itiinoaio (rwrriiffMSm. 

This is the highly important problem which Pythagoras is credited with 
having solved. Compare the passage from Plutarch {Symp. vtu. 2, 4) quoted 
■n the note on i. 44 above, Vol, 1. pp. 343 — 4. 

We are bidden to construct a rectilineal figure which shall have the form of 
one and the siu of another rectilineal figure. The corresponding proposition 
of the Data, Prop. 55, asserts that, "if an area {xtapiav) be given in form 
{(iSii) and in magnitude, its sides will also be given in magnitude." 

Simson sees signs of corruption in the text of this proposition also. In 
the first plac^ the proof speaks of the triangle ABC, though, according to the 
enunciation, the figtire for which ABC is taken may be any rectilineal figure, 
tutfuypafifiop "rectilineal figure" would be more correct, or <ISo9, "figure"; the 
mistsjie, however, of using rpiyavoy is not one of great importance, being no 



VI. »5. »6] PROPOSITIONS 25, a6 »S5 

doubt due to the accident by which the tigure was drawn as a. triangle in the 
diagram. 

The other observation is more important. After Euclid has proved that 

(fig. ABC) ; (fig. JCG/T) = {B£) : {Ef), 

he might have inferred directly from v. 14 that, since ABC is equal to BE, 
KGH is e<jual to EF. For v. 14 includes the proof of the fact that, if A is 
to .^ as C IS to D, and A is etjual to C, then B is equal to D, or that of four 
proportional magnitudes, if the first is equal to the third, the second is equal 
to the fourth. Instead of proceeding in this way, Euclid first permutes the 
proportion by v. 16 into 

(fig. ABC) : {BE) = (fig. KGH) : (EF), 

arid then infers, as if the inference were easier in this form, that, since the 
Jlftt is equal to the secend, the third is equal to the fourth. Yet there is no 
proposition to this effect in Euclid. The same unnecessary step of permutation 
is also found in the Greek text of xi. 23 and xii. 2, 5, ir, 12 and 18. In 
reproducing the proofs we may simply leave out the steps and refer to v. 14. 



Proposition 26. 

If frotn a parallelogram there be taken away a parallelo- 
gram similar and mnilarly situated to the whole and having 
a common angle with it, it is about the same diameter with the 
whole 

For from the parallelogram A BCD let there be taken 
away the parallelogram AF similar and 
similarly situated to ABCD, and having 
the angle DAB common with it ; 

I say that ABCD is about the same 
diameter with AF. 

For suppose it is not, but, if possible, 
let AHCh& the diameter < oiABCD > , 
let GF be produced and carried through 
to ff, and let /fK be drawn through // 
parallel to either of the straight lines AD, BC. [i. 31] 

Since, then, ABCD is about the same diameter with KG, 
therefore, as DA is to AB, so is GA to AK. [vi. 24] 

But also, because of the similarity of ABCD, EG, 

as DA is to AB, so is GA to AE ; 

therefore also, as GA is to AK, so is GA to AE. [v. 1 1] 

Therefore GA has the same ratio to each of the straight 
lines AK, AE. 




•S« •> BOOK VI [VI. !t6 

Therefore A£ is equal to ^A' [v. 9], the less to the 
greater: which is impossible. 

Therefore A BCD cannot but be about the same diameter 
with AF\ 

therefore the parallelogram A BCD is about the same diameter 
with the parallelogram AF. 

Therefore etc. 

Q. E. D. 

" For suppose it is not, but, if possible, let AHC be the diameter." WKat 
is meant is " For, if AFC is not the diameter of the paralleiogram AC, let 
AlfC be its diameter." The Greelt text has fo-Tut amav Sta/wrpiK ij AOT; 
but clearly avrwv is wrong, as we cannot assume that one straight line is the 
diameter of both parallelograms, which is just what we have to prove. F and 
V omit the auTiui-, and Heiberg prefers this correction to substituting aurou 
after Peyrard. I have inserted " < of A BCD > " to make the meaning clear. 

If the straight line AHC does not pass through F, it must meet either 
GF (X £?/' produced in some point H. The reading in the text "and let 
GF be produced and carried through io H " (uat im^X-ifivxta, ^ HZ &7xSm Jjrt 
TO ®) corresponds to the supposition that H is on GF produced. The words 
were left out by Theon, evidently because in the figure of the mss. the letters 
E, Z and K, were interchanged, Heiberg therefore, following August, has 
preferred to reKiin the words and to correct the figure, as well as the passage in 
the text where AE, AK were interchanged to be in accord with the MS. figure. 

It is of course possible to prove the proposition directly, as is done by 
Dr Lachlan. Let AF, ACh^ the diagonals, and let us matte no assumption 
as to how they fall. 

Then, since EF\s parallel to AG and therefore to BC, 

the angles AEF, ABC are equal. c 

And, since the paraRetograms are similar, 

AE : EF=AB : BC. [vi. Def. i] 

Hence the triangles AEF, ABC are similar, [vi. 6] 

and therefore the angle FAE is equal to the angle CAB. 

Therefore AFfalh on AC. 

The proposition is equally true if the parallelogram which is similar and 
similarly situated to the given parallelogram is not " taken 
away" from it, but is so placed that it is entirely outside the 
other, while two sides form an angle vertically opposite to 
an angle of the other. In this case the diameters are not 
"the same," in the words of the enimciation, but are in 
a straight line with one another. This extension of the 
proposition is, as will be seen, necessary for obtaining, 
according to the method adopted by Euclid in his solu- 
tion of the problem in vi. s8, the second solution of that 
problem. 





vj. a?} PROPOSITIONS 26, 27 



Proposition 27. 

0/ all ike parallelograms applied to the same straight line 
and deficient by parallelogrammic figures similar and similarly 
situated to that described on the half of the straight line, that 
parallelogram is greatest which is applied to the half of the 
straight line and is similar to the defect. 

Let AB be a straight line and let it be bisected at C; 
let there be applied to the straight 
line AB the parallelogram AD 
deficient by the parallelogrammic 
figure DB described on the half of 
AB, that is, CB; 

I say that, of all the parallelograms 
applied to AB and deficient by 
parallelogrammic figures similar and 
similarly situated to T)B, A J? is greatest. 

For let there be applied to the straight line AB the 
parallelogram AF deficient by the parallelogrammic figure 
FB similar and similarly situated to DB; 
I say that AD is greater than AF. 

P'or, since the parallelogram DB is similar to the parallelo- 
gram FB, 

they are about the same diameter. [vi. 36] 

Let their diameter DB be drawn, and let the figure be 
described. 

Then, since CF is equal to F£, _ [i. 43] 

and FB is common, 

therefore the whole C// is equal to the whole /CF. 

But C// is equal to CG, since AC is also equal to CB, 

[I. 36] 
Therefore GC is also equal to F/l. 
Let CF be added to each ; 

therefore the whole AFis equal to the gnomon LMN\ 

so that the parallelogram DB, that is, AD, is greater than 
the parallelogram AF. 

Therefore etc. 



as* BOOK VI [vi. aj 

We have already (note on i. 44) seen the significance, in Greek geometry, 
of the theory of " the application of areas, their exceeding and their falling- 
short." In I. 44 it was a question of " applying to a given straight line 
(exactly, without 'excess' of 'defect') a parallelogram equal to a given 
rectilineal figure, in a given angle," Here, in vi. 27 — 29, it is a question 
of parallelograms applied to a straight line but "deficient (or exceeding) by 
paralUiograms similar aiid similarly 
situated to a given parallelogram." \^ 

Apart from size, it is easy to construct ^ 7 

any number of parallelograms " de- r / '^^ 7" 

ficient" or "exceeding" in the manner / / / ^\A l 

described. Given the straight line /' e F ^(™...... 

AB to which the parallelogram has to L . ('....^^s/ 

be applied, we describe on the base ^ ^ '^\ 

CB, where C is on AB, or on BA 

produced beyond A, any parallelogram " similarly situated " and either equal 
or similar to the given parallelogram (Euclid takes the similar and similarly 
situated parallelogram on half the line), draw the diagonal BD, take on it 
(produced if necessary) an> points as E, K, draw EF, or KL, parallel to CI) 
to ineet AB ot AB produced and complete the parallelograms, as AH, ML, 

If the point E is taken on BD or BD produced beyond Z>, it must be so 
taken that EF meets AB between A and B. Otherwise the parallelogram 
AE would not be applied to AB itself, as it is required to be. 

The parallelograms BD, BE, being about the same diameter, are similar 
[vi, 24], and BE is the defect of the parallelogram AE relatively to AB, 
AE is then a parallelogram applied ta AB but deficient by a parallelogram 
similar and similarly situated to BD, 

If A* is on D3 produced, the parallelogram BK is similar to BD, but it 
is the excess of the parallelogram AK relatively to the base AB. AK h a. 
parallelc^ram applied to AB but exceeding by a parallelogram similar and 
similarly situated to BD. 

Thus it is seen that BD produced both ways is the locus of points, such 
9S E OT K, which determine, with the direction of CD, the position of A, and 
the direction of AB, parallelograms applied to AB and deficient or exceeding 
by parallelograms similar and similarly situated to the given parallelogram. 

The importance of vi. 27 — 29 from a historical point of view cannot be 
overrated. They give the geometrical equivalent of the algebraical solution 
of the most general form of quadratic equation when that equation has a real 
and positive root. It will also enable us to find a real negative root of a 
quadratic equation ; for such an equation can, by altering the sign of x, be 
turned into another with a real positive root, when the geometrical method 
again becomes applicable. It will also, as we shall see, enable us to represent 
both roots when both are real and positive, and therefore to represent both 
roots when both are real but either positive or negative. 

The method of these propositions was constantly used by the Greek 
geometers in the solution of problems, and they constitute the foundation of 
Book X. of the Elements and of ApoUonius' treatment of the conic sections. 
Simsons observation on the subject is entirely justified. He says namely on 
VI. 28, 29: "These two problems, to the first of which the 27th Prop, is 
necessary, are the most general and useful of all in the Elements, and are 
most frequently made use of by the ancient geometers in the solution of 
other problems ; and tbereft)re are very ignorantly left out by Tacquet and 



VI. aj] PROPOSITION 27 959 

Dechales in their editions of the Elements, who pretend that they are scarce 
of any use." 

It is strange that, with this observation before him, even Todhunter should 
have written as follows. " We have omitted in the sixth Book Propositions 
37, z8, 39 and the first solution which Euclid gives of Proposition 30, as they 
appear now to be never required, and have been condemned as useless t^ 
various modern commentators ; see Austin, Walker and Lardner." 

VI. 27 contains the Stapto-ftat, the condition for a real solution, of the 
problem contained in the proposition following it The maximum of a!l the 
parallelograms having the given property which can be applied to a given 
straight line is that which is described upon half the line (to airo t^s ^(luntat 
ayaypa^/urmr). This corresponds to the condition that an equation of the 
form 

u'^ -/jc* = A 

may have a real root. The correctness of the result may be seen by taking 
the case in which the parallelograms are 
rectangles, which enables us to leave out 
of account the sint of the angle of the 
parallelograms without any real loss of 
generality. Suppose the sides of the rect- 
angle to which the dtfeci is to be similar 
to be as £ to f, ^ corresponding to the 
side of the defect which lies along AB. 
Suppose that AKFG is any parallelogram 

applied to AB having the given property, that AB=a, and that JFK=x. 
Then 

KB = -x, and therefore AK=a- - x. 

c ( 

Hence [a-- x\x=-S, where S is the area of the rectangle AKFG. 
Thus, given the equation 



F 

X 



C K 



c 






where 5 is undetermined, vi. J7 tells us that, if x is to have a real value, S 
cannot be greater than the rectangle CE, 

Now CB = -, and therefore CD = -..-; 
a' b 2' 

whence 5 It i . — , 

which is just the same result as we obtain by the algebraical method. 

In the particular case where the defect of the parallelogram is to be t 
square, (he condition becomes the statement of the fact that, if a straight lint 
be divided into two parts, the rectangle contained by the parts cannot exceed the 
squart on half the line. 

Now suppose that, instead of taking F on BD as in the 6gure of the 
proposition, we take F on BD produced beyond D but so that DF is less 
than BD. 

Complete the figure, as shown, after the manner of the construction in 
the proposition. 




afitt) • BOOK VI [vi. 37, 18 

Then the pamllelogram FKBH is similar to the given paratlelogram to 
which the defect is to be simikr. Hence the parallelogram GAKF is also a 
paralletograin applied to AB and satisfying 
the given condition. 

We can now prove that GAKF is less 
than CE or AD. 

Let ED produced meet AG in O. 

Now, since BF is the diagonal of the 
parallelogram KH, the complements KD, 
DH are equal. 

But 

DH= DG, and DG is greater than OF. 
Therefore KD > OF. 

Add OK to each ; 
and AD, or CE,>AF, • • "■' 

This other "" case " of the proposition is found in all the MSS,, but Heiberg 
relegates it to the Appendix as being very obviously interpolated. The 
reasons for this course are that it is not in Euclid's manner to give a separate 
demonstration of such a " case "; it is rather his habit to give one case only 
and to leave the student to satisfy himself about any others (cf. i, 7). Internal 
evidence is also against the genuineness of the separate proof. It is put after 
the conclusion of the proposition instead of before it, and, if Euclid had intended 
to discuss two cases, ne would have distinguished them at the beginning of 
the proposition, as it was his invariable practice to do. Moreover the second 
"case" is the less worth giving because it can be so easily reduced to the 
first. For suppose F' to be taken on BD so that FD - F D. Produce BF 
to meet AG produced in P. Complete the parallelogram BAPQ, and draw 
through F" straight lines parallel to and meeting its opposite sides. 

Then the complement F'Q is equal to the complement AF'. 

And it is at once seen that AF, F'Q are equal and similar. Hence the 
solution of the problem represented by AF or F'Q gives a parallelogram of 
the same size as AF" arrived at as in the first " case." 

It is worth noting that the actual difference between the parallelogram 
AF and the maximum area AD that it can possibly have is represented in 
the figure. The difference is the small parallelc^ram DF. 

Proposition 28. 

To a given straight line to apply & parallelogram equal to 
a given rectilineal figure and deficient by a parallelogrammic 
figure similar to a. given one : thus the given rectilineal figure 
must not be greater than the parallelogram, described on the 
half of the straight line and similar to the defect. 

Let AB be the given straight line, C the given rectilineal 
figure to which the figure to be applied to AB is required to 
be equal, not being greater than the parallelogram described 
on the half of AB and similar to the defect, and D the 
parallelogram to which the defect is rt;quired to be similar ; 



VI. 28] PROPOSITIONS 17, 28 i6t 

thus it is required to apply to the given straig^^ line AB a 
parallelogram equal to the given rectilineal figure C and 
deficient by a parallelogrammic figure which is similar to D, 

Let ABhe bisected at the point E, and on EB let EBFG 
be described similar and similarly situated to D ; [vi. 18] 

let the parallelogram AG he completed. 

If then AG IS equal to C, that which was enjoined will 
have been done ; 

for there has been applied to the given straight line AB 
the parallelogram AG equal to the given rectilineal figure C 
and deficient by a parallelogrammic figure GB which is similar 
to/?. 




But, if not, let //E be greater than C, 
Now HE is equal to GB ; 

therefore GB is also greater than C, 

Let KLMN be constructed at once equal to the excess 
by which GB is greater than C and similar and similarly 
situated to D. [vi, 25} 

But D is similar to GB \ 
therefore KM \s also similar to GB. [vi. zi"] 

Let, then, KL correspond to GE, and LM to GF. 
Now, since GB is equal to C, KM, 

therefore GB is greater than KM ; 
therefore also GE is greater than KL, and GF than LM. 

Let GO be made equal to KL, and GP equal to LM; 
and let the parallelogram OCPQ be completed ; 
therefore it is equal and similar to KM. 

Therefore GQ is also similar to GB; \y\. i\\ 

therefore GQ is about the same diameter with GB. [vi. a6] 



afo r BOOK VI .;t't [vi. aS 

,. Let C55 be their diameter, and let the figure be described. 
Then, since BG is equal to C, KM, 

and in them GQ is equal to KM, 

therefore the remainder, the gnomon UWV, is equal to tht 

remainder C. 

And, since PR is equal to OS, ' '' 

let QB L e added to each ; ' • ••* ' • 
therefore the whole PB is equal to the whole OB. 

But OB is equal to TE, since the side AE is also equal 
to the side EB ; [1. 3^] 

therefore TE is also equal to PB. 

Let 05 be added to each ; 

therefore the whole TS is equal to the whole, the gnomon 
VWU. 

But the gnomon VWU was proved equal to C ; 
therefore TS is also equal to C. 

Therefore to the given straight line AB there has been 
applied the parallelogram ST equal to the given rectilineal 
figure Cand deficient by a parallelogrammic figure ^^ which 
is similar to D. 

Q. E. F, 



The second part of the enunciation of this proposition which states the 
SiapuT/uH appears to have been considerably amplified, but not improved in 
the process, by Theon. His version would read as follows. " But the given 
rectilineal figure, that namely to which the applied parallelogmm must be 
equal (tf S«t lo-ov impa^aXtiv), must not be greater than that applied to the half 
(irapa/3iiAAo^(Miv instead of avaypa^/inov), the defects being similar, (namely) 
that (of the parallelogram applied) to the half and that (of the required 
parallelogram) which must have a similar defect" {onoimr otrui' tuv IWtt/i- 
fidrwv roD rt air^ rij^ ^fjiurtvK koI cp Sft uftoiov iXKtiTrnv). The first ampHflCfttion 
"that to which the applied parallelogram must be equal" is quite unnecessary, 
since " the given rectilineal figure " could mean nothing else. The above 
attempt at a translation will show how difficult it is to make sense of the 
words at the end , they speak of two defects apparently and, while one may 
well be the " defect on the half," the other can hardly be tfu pvm paralUia^am 
" to which the defect (of the required [^rallelogram) must be similar." Clearly 
the reading given above (from P) is by far the better. 

In this proposition and the next there occurs the tacit assumption (already 
alluded to in the note on vi. 31) that if, of two similar paraiitlogramz, one is 
gnatfr than the of her, ttther side of the greater is greater tkan the corresponding 
side of the kit. 



VI. 98] 



PROPOSITION sS 



•*3 



As already remarked, vi. 28 is the geometrical equivalent of the solution 
of the quadratic equation 



tf jc - - a* • 

c 



S. 



subject to the condition necessary to admit of a real solution, namely that 

* 4 

The corresponding proposition in the Data is (Prop. 58), If a given {ana) 
be applied (i.e. in the fonn of a parallelogram) to a given straight line and be 
defieitnt by a figure (i.e. a parallelogram) f<Wfl in species, the breadths of tht 
defeet are given. 

To exhibit the exact correspondence between Euclid's geometrical and 
the ordinary algebraical method of solving the equation we will, as before 
(in order to avoid bringing in a constant dependent on the sine of the angle 
of the parallelograms), suppose the parallelograms to be rectangles. To solve 
the equation algebraically we change the signs and write it 

-x'-ax = -S. 

e a* 



We may now complete the square by adding 



*'4' 



Thus 



■<Mf + T. — - 

b 4 



e a^ „ 



and, extracting the square root, we have 

Now let us observe Euclid's method. - ^ 



o \ 

S 

X 




s a 



■0 



He first describes GEBF on EB (half of AB") similar to the given 
parallelogram D. 

He then places in one angle FGE of GEBF a similar and similarly 
situated parallelogram GQ, equal to the difference between the parallelogram 
GB and the «irea C. 



With our notation, 
whence 



GO: OQ = e:b, 



» 



OQ = GO.'^ 



a64 BOOK VI [vi. 38 

Similarly -^ = £B=GE.-, 

so that GE=i.-. ' 

a 

Therefore the parallelogram GQ^ GO* . - , 

( a' 
and the parallelogram GB = 1 • — • 

Thus, in taking the parallelogram GQ equal to {GB - S), Euclid really 
finds GO from the equation 

GO'.''-^i.''--S. 

f * 4 

The value which he finds is ^ 

and he finds QS (or x) by subiracting GO from GE ; whence 
^_t a f'cfe a«^\ 

It will be observed that Euclid only gives one solution, that corresponding 
to the negative sign before the radical. But the reason must be the same as that 
for which he only gives one "case" in vi. s 7. He cannot have failed to see how 
to adti GO to GE would give another solution. As shown under the last 
proposition, the other solution can be arrived at 

(i) by placing the parallelogram GOQ/' in B' A' 

the angle vertically opposite to FGE so that '. ''--.Q' ^ 

G(^ lies along BG produced. The parallelo- \ Yx ^' ; \ 
gram A Q then gives the second solution. The \ I \^ \ \ \ 

side of this parallelogram lying along AB \% \ \p' x\ '. ' 

equal to S3. The other side is what we have \ T ' gVv p\ \ 
called Xy and in this case \ \ \ \^ ', \ 

x^EG+GO J \ o\ \iQ\ 



' * ■ 2 ^ V i (i • 4 -^j ■ 



S B 



(z) A parallelogram similar and equal to A<^ can also be obtained by 
producing BG till it meets A T produced and completing the parallelogram 
JfABA', whence it is seen that the complement QA' is equal to the comple- 
ment j4^, besides being equal and sin'ilar and similarly situated to AQ. 

A particular case of this proposition, indicated in Prop. 85 of the Data, is 
that in which the sides of the defect are equal, so that the defect is a rhombus 
with a given angle. Prop. 85 proves that, 1/ two straight lints (oniain a 
givtn area in a given angle, and the sum 

of the straight lines be given, each of them E *_ 

will be gioen also. AB, BC being the / 

given straight lines "containing a given / 

area AC in ^ given angle ABC," one / 

side CB is produced to J) so that BD ^ 

is equal to AB^ and the parallelograms are 

completed. Then, by hypothesis, CD is of given length, and .^C is a parallelo- 



VI. a8, 29] PROPOSITIONS 28, 19 m^; 

gram applied to CZ> falling short by a rhombus {AD) with a given angle 
EDB, The case is thus a particular case of Prop. 58 of the Data quoted 
above (p. 263) as corresponding to vi. 28. 

A particular case of the last, that namely in which the defect \& a squart, 
corresponding to the equation 

is importai.t. This is the problem of applying to a gwen straight lint a 
tedangli tqual to a given area and falling short by a square ; and it can be 
solved, without the aid of Book vi., as shown above under 11. 5 (Vol. 1. 
PP- 383—4). 



Proposition 29. 

To a given siraigkt line to apply a parallelogram equal to 
a given rectilineal figure and exceeding by a parallelogrammic 
figure similar to a given one. 

Lei j4B be the given straight line, C the given rectilineal 
figure to whtcli the figure to be applied to A J} is required to 
be equal, and D that to which the excess is required to be 

similar ; 

thus it is required to apply to the straight line AB a parallelo- 
gram equal to the rectilineal figure C and exceeding by a 
parallelogrammic figure similar to D. 




Let AB be bisected at ^ ; , 

let there be described on BB the parallelogram B/^ similar 
and similarly situated to D ; 

and let G// be constructed at once equal to the sum of BB", 
C and similar and similarly situated to £f. [vi. 35] 

Let JC// correspond to BL and /CG to BB. 
Now, since G// is greater than BB, 

therefore J^// is also greater than BL, and JCG than BB. 



SH BOOK VI [vi. 39 

Let PL, FE be produced, 
let FLM be equal to KH, and FEN to KG, 
and let MN be completed ; 

therefore MN is both equal and similar to GH, 
But GH is similar to EL ; 
therefore ^A?" is also similar to EL ; [vi, a i] 

therefore J?/, is about the same diameter with MN. [vi. 16] 
Let their diameter FO be drawn, and let the figure be 
described. 

Since GH'\s equal to EL, C, 
while GH is equal to MN, 
therefore MN is also equal to EL, C. 
Let EL be subtracted from each ; 
therefore the remainder, the gnomon XIVV, is equal to C. 
Now, since AE is equal to EB, 
AN is also equal to NB [1. 36], that is, to LP [i. 43]- 
Let EO be added to each ; 

therefore the whole AO is equal to the gnomon VIVX, 
But the gnomon VJVX is equal to C \ 

therefore AO is also equal to C. 
Therefore to the given straight line AB there has been 
applied the parallelogram AO equal to the given rectilineal 
figure C and exceeding by a parallelogram mic figure QP 
which is similar to D, since PQ is also similar to EL [vi. n]. 

Q. E. F. 

The corresponding proposition in the Data is (Prop. 59), If a given {area) 
Ar applied (i.e. in the form of a parallelogram) tc a given straight lint exteeding 
by a fi^re gii'en in species, the breadths of the excess are givi». 

The problem of vi, 29 corresponds of course to the solution of the 
quadratic equation 

ax -y- - 3? = S, 
c 



The algebraical solution of this equation gives 



The exact correspondence of Euclid's method to the aigebraical solution 
may be seen, as in the case of vi. 28, by supposing the parallelograms to be 
rectangles. In this case Euclid's construction on EB of the parallelogram 
EL similar to D is equivalent to finding that 

FE = \.^-, and EL = \.'^. 
b z * 4 



VI. 19. 3°} PROPOSITIONS 19, 30 *&} 

His determination of the similar parailelogram />/jV equal to the sum of £L 
and S corresponds to proving that 

f d 4 






or 

whence x is found as 

a 

Euclid takes, in this case, the solution corresponding to the positive sign 
before the radical because, from his point of view, that would be the only 
solution. 

No iiofiitriiai is necessary because a real geometrical solution is always 
possible whatever be the size of S. 

Again the £>afa has a proposition indicating the particular case in which 
the excess is a rhombus with a given angle. Prop. 84 proves that, 1/ itvo 
straight lines contain a givtn area in a ^ven angle, and one of thf straight lines 
is greater than the other by a given straight line, each of the two straight lines is 
given also. The proof reduces the proposition to a particular case of Data, 
Prop. 59, quoted above aa corresponding to vi. 29. 

Again there is an important particular case which can be solved by means 
of Book II. only, as shown under 11. 6 above {Vol. 1. pp. 386 — 8), the case namely 
in which the excess is a square, corresponding to the solution of the equation 

This is the problem of applying to a given straight line a rectangle equal to a 
ffven area and exceeding by a square. 



Proposition 30. 

To cut a given finite straight line in extreme and mean 
ratio. 

Let AB be the given finite straight line ; 

thus it is required to cut AB in extreme and mean ratio. 

On AB let the square BC be described ; 
and let there be applied to AC the parallelo- 
gram CD equal to BC and exceeding by 
the figure AD similar to BC. [vi. 29] 

Now BC is a square ; 
therefore AD is also a square. 

And, since BC is equal to CD, 

let CE be subtracted from each ; 

therefore the remainder BF is equal to 
the remainder AD. 




•88 cf , BOOK VI [vi. 30, 31 

But it is also equiangular with it ; 
therefore in BF, AD the sides about the equal angles are 
reciprocally proportional ; [vi- h] 

therefore, as FE is to ED, so is AE to EB, 
But FE is equal to AB, and ED to AE, 
Therefore, as BA is to AE, so is AE to EB. 

And AB is greater than AE ; 

therefore AE is also greater than EB. 
Therefore the straight line AB has been cut in extreme 
and mean ratio at E, and the greater segment of it is AE. 

Q. E. F. . 

It will be observed that the construction in the tent is a direct application 
of the preceding Prop. 29 in the particular case where the txass of the 
parallel ogram which is applied is a squar*. This fact coupled with the 
position of V(, 30 is a sufficient indication that the construction is Euclid's. 

In one place Theon appears to have amplified the argument. The text 
above says "But fE is equal to AB" while the mss. B, F, V and p have 
" But /^ff is equal to ^ C, that is, to ^^." 

The MSS. give after ^vtp Ihu iroujiriH an alternative construction which 
Heibei^ relegates to the Appendix. Thf text-books give this construction 
alone and leave out the other. It will be remembered that the alternative 
proof does no more than refer to the equivalent construction in ii. 11. 

"Let AB \ie. cut at C so that the rectangle AB, BC is equal to the 
square on CA. [ii. ( jj 

Since then the rectangle AB, BC is equal to the stjuare on CA, 
therefore, as BA is to .rfC, so is /4Cto CB. [vi. 17] 

Therefore AB has been cut in extreme and mean ratio at tV 

It is intrinsically improbable that this alternative construction was added 
to the other by Euclid himself. It is however just the kind of interpolation 
that might be expected from an editor. If Euclid had preferred the alternative 
construction, he would have been more likely to give it alone. 

I '■ 

Proposition 31. 

In right-angled triangles the figure on Ike side suit ending 
the right angle is equal to the similar and similarly described 
Jtgures on the sides containing the right angle. 

L&t ABC be a right-angled triangle having the angle BAC 
right ; 

I say that the figure on BC is equal to the similar and 
similarly described figures on BA, AC. 

Let AD be drawn perpendicular. 

Then since, in the right-angled triangle ABC, AD has 



Vf. 3'] 



PROPOSITIONS 30, 31 




been drawn from the right angle at A perpendicular to the 
base BC, 

the triangles ABD, v4/?C adjoin- 
ing the perpendicular are similar 
both to the whole ABC and to 
one another, [vi, 8] 

And, since ABC is similar to 
ABD, 

therefore, as CB is to BA^ so is 
AB to BD. [vi. I>er. 1] 

And, since three straight lines 
are proportional, 

as the first is to the third, so is the figure on the first to the 
similar and similarly described figure on the second, [vi. 19, For.] 

Ihercfore, as CB is to BD, so is the figure on CB to the 
similar and similarly described figure on BA. .-•■• ■ ^ 'l^ '. 

For the same reason also, ' 

as BC is to CD, so is the figure on BC to that on CA ; 
so that, in addition, 

as BC is to BD, DC, so is the figure on BC to the similar 
and similarly described figures on BA, AC. 

But BC is equal to BD, DC ; 
therefore the figure on BC is also equal to the similar and 
similarly described figures on BA, AC. 

Therefore etc. 

Q. E. D. 

As we have seen (note on i. 47), this extension of [. 4; is credited by 
Proclus to Euclid persor\ally. 

There is one inference in the proof which requires examination. Euchd 
proves that 

CB : .ff/> = (figure on CB) -. (figure on BA), 

and that BC : CZ> = (figure on BC) : (figure on CA), 

and then infers directly that 

BC : {BD+ CZ))- (fig. on BC) : (sum of figs, on BA and AC). 

Apparently v. 24 must be relied on ss justifying this inference. But it is not 
directly applicable ; for what it proves is that, if 

a:b = c:d, .,-..• A-^ >.V-. 

and ■ f:b=f:d, • .-'M^V 

then (a + (t) : b = {f \f) : d. 

Thus we should itwtrt the first two proportions given above (by Simson's 



»jo «t . BOOK VI [vi. 31, 31 

Prop. B which, as we have seen, is a direct consequence of the definition of 
proportion), and thence infer by v. 34 that 

{BD+CD) : J3C= {sum of figs, on £j4, AC) : (fig. on BC). 
But BD 4 CD is equal to BC; 

therefore (by Simson's Prop. A, which again is an immediate consequence of 
the definition of proportion) the sum of the figures on BA, AC is equal to 
the figure on BC. 

The Mss. ag^in give an alternative proof which Heibetg places in the 
Appendix. It first shows that the simitar figures on the three sides have the 
same ratios to one another as the squara on the sides respectively. Whence, 
by using I. 47 and the same argument based on v. 24 as that explained above, 
the result is obtained. 

If it is considered essential to have a proof which does not use Simson's 
Props, fi and A or any proposition but those actually given by Euclid, no 
method occurs to me except the following. 

Eucl. V. 12 proves that, if a, J, c are three magnitudes, and d, e, f three 
Others, such that 

a : b=:d : t, 
• •..-:- :.: ■ bie'-f.f, • ., : 

then, ex aeptali, a : c = d :/, , ' i ' 

If now in addition a •.b=b -.Cf 

so that, also, d i e = e :/, 

the ratio a -. ( ii duplicate of the ratio a : i, and the ratio d :/ duplicate of 
the ratio d : t, whence the ratios which ar< duplicate of equal ratios are equal. 
Now (fig. on AC)\ (fig. on AB) = the ratio duplicate of AC : AB 

= the ratio duplicate of CD : DA 
= CD : BD. 
H»ice (sum of figs, on AC, AB) : (fig. on AB) = BC -. BD. [v. iS] 
But ' (fig. on BC) : (fig. on AB) = BC : BD 

(as in Euclid's proof). 
Therefore the sum of the figures on AC, AB has to the figure on AB the 
same ratio as the figure on BC has to the figure on AB, whence 

the figures on AC, AB are together equal to the figure on BC- [v. 9] 



Proposition 32. 

If two triangles having two sides proportional to two sides 
be placed together at one angle so that their corresponding sides 
are also parallel, the remaining sides of the triangles will be 
in a straight line. 

Let ABC, DCE be two triangles having the two sides 
BA, AC proportional to the two sides DC, DE, so that, as 
AB is to A C, so is DC to D£, and AB parallel to DC, and 
AC to D£; 
I say that BC is in a straight line with C£. 




yi. 3a] PROPOSITIONS 31, 3a ^1 

For, since AB is parallel to DC, 

and the straight line AC has fallen upon them, 

the alternate angles BAC, ACD 
are equal to one another, [r. 19] 

For the same reason 

the angle CDE is also 
equal to the angle A CD ; 
so that the angle BAC is equal 
to the angle CDE. 

And, since ABC, DCE are 
two triangles having one angle, the angle at A, equal to one 
angle, the angle at D, 

and the sides about the equal angles proportional, 
so that, as BA \%\ja AC, so is CD to DE, 

therefore the triangle ABC is equiangular with the 
triangle DCE ; [vi. 6] 

therefore the angle ABC is equal to the angle DCE. 
But the angle ^CZ? was also proved equal to the angle 
BAC\ 

therefore the whole angle A CE is equal to the two angles 
ABC, BAC. 

Let the angle ACB be added to each ; 

therefore the angles ACE, ACB are equal to the angles BAC, 
ACB, CBA. 

But the angles BA C, ABC, A CB are equal to two right 
angles ; [1. 3a] 

therefore the angles ACE, ACB are also equal to two 
right angles. 

Therefore with a straight line AC, and at the point C on 
it, the two straight lines BC, CE not lying on the same side 
make the adjacent angles ACE, ACB equal to two right 
angles ; 

therefore BC is in a straight line with CE. [1. 14] 

Therefore etc. 

Q. E. D. 

It has often been pointed out (e.g. by Clavm&, Lardner and Todhunter) 
that the enunciation of this proposition is not precise enough. Suppose that 




ijt BOOK VI ' [vi. 32 

ABC is a triangle. From C draw CD parallel to BA and of any length. 
From D draw D£ parallel to CA and of such length that 

CD:DE = BA : AC. 
Then the triangles ABC, BCD, which have the angular jxiint C coniRion 
literally satisfy Euclid's enunciation ; but by no possi- 
bility can CE be iti a straight line with CB if, as 
in the case supposed, the angles included by the 
corresponding sides are supplementary (unless both are 
right angles). Hence the included angles must be 
equal, so that the triangles must be similar. That 
being so, if they are £0 have nothing more than one 
angular point common, and two pairs of corresponding 
sides are to he/aralMas distinguished froni one or both being in the same 
slraigiU line, the triangles can only be placed so that the corresponding sides 
in both are on the same side of the third side of either, and the sides (other 
than the third sides) which meet at the common angular point are not corre- 
sponding sides. 

Todhunter remarks that the proposition seems of no use. Presumably he 
did not know that it is used by Euclid himself in xiii. 17. This is so 
however, and therefore it was not necessary, as several writers have thought, to 
do away with the proposition and Irnd a substitute which should be more useful. 

1. De Morgan proposes this theorem : "If two similar triangles be placed 
with their bases parallel, and the equal angles at the bases towards the same 
parts, the other sides are parallel, each to each ; or one pair of sides are in 
the same straight line and the other pair are parallel," 

2. Ur Lachlan substitutes the somewhat similar theorem, "If two similar 
triangles be placed so that two sides of 

the one are parallel to the corresponding q 

sides of the other, the third sides are a /\ 

parallel," /\ / \ 

But it is to be observed that these qZ.....,\.. Z. a 

propositions can be proved without / \ 

using Book vi. at all ; they can be / \ 

proved from Book i,, and the triangles = c 

may as well be called "equiangular" 

simply. It is true that Book vi. is no more than formally nece^ary to 

Euchd's proposition. He merely uses vt, 6 because his enunciation does not 

say that the triangles are similar ; and he only proves them to be similar in 

order to conclude that they are equiangular. From this point of view 

Mr Taylor's substitute seems the best, viz, 

3. "If two triangles have sides parallel in pairs, the straight lines joining 
the corresponding vertices meet in a point, 

or are parallel." 

Simson has a theory (unnecessary in 
the circumstances) as to the possible 
object of VI. 32 as it stands. He points 
out that the enunciation of vi, 26 might 
be more general so as to cover the case 
of similar and similarly situated parallelo- 
grams with equal angles not coincident 
but vertically opposite. It can then be proved that the diagonals drawn 




VI- 3S. 33] 



PROPOSITIONS 32, 33 



ns 



through the common angular point are In one straight line. If ABCF, CDEG 
be similar and simikrl)' situated parallelograms, 
so that BCGf DCF are straight lines, and if 
the diagonals AC^ CE be drawn, the triangles 
ABC, CDR are similar and are plated exactly 
as deseriied in vi, 32, so that AQ CE are in a 
straight line. Hence Sim son suggests that 
there may have been, in addition to the in- 
direct demonstration in vi. 26, a direct proof 
covering the case just given which may have 
used the result of vi, 31. I think however 
that the place given to the latter proposition in Book vi. is against this view. 




Proposition 33. 

In equal circles angles have the same ratio as the circum- 
ferences on which they stand, whether they stand at the centres 
or at the circumferences. 

Let ABC, DEF be equal circles, and let the angles BGC, 
EHFhG. angles at their centres G, H, and the angles BAC, 
£'/?/^ angles at the circumferences ; 

I say that, as the circumference BC is to the circumference 
£F, so is the angle BGC to the angle EHF, and the angle 
^^C to the angle jp/?/^. 




For let any number of consecutive circumferences CK, 
KL be made equal to the circumference BC, 
and any number of consecutive circumferences FM, MN equal 
to the circumference EF', 
and let GK, GL, HM, HN be joined. 

Then, since the circumferences BC, CK, KL are equal 
to one another, 

the angles BGC, CGK, KGL are also equal to one another ; 

[in. 27] 



i14 BOOK VI ' [vi. 33 

therefore, whatever multiple the circumference BL is of BC, 
that multiple also is the angle BGL of the angle BGC. 

For the same reason also, 
whatever multiple the circumference NE is of EF, that 
multiple also is the angle NHE of the angle EHF. 

If then the circumference BL is equal to the circumference 
^A^, the angle BGL is also equal to the ^ug\&EHN; [in. ij] 
if the circumference BL is greater than the circumference 
EN, the angle BGL is also greater than the angle EHN ; 
and, if less, less. 

There being then four magnitudes, two circumferences 
BC, EF, and two angles BGC, EHF, 

there have been taken, of the circumference BC and the angle 
BGC equimultiples, namely the circumference BL and the 
angle BGL, 

and of the circumference EF and the angle EHF equi- 
multiples, namely the circumference EN and the angle EHN. 

And it has been proved that, 
if the circumference BL is in excess of the circumference EN, 
the angle BGL is also in excess of the angle EHN ; 
if equal, equal ; 
and if less, less. 

Therefore, as the circumference BC is to EF, so is the 
angle BGC to the angle EHF. [v. Def. 5] 

But, as the angle BGC is to the angle EHF, so is the 
angle BAC to the angle EDF; for they are doubles respec- 
tively. 

Therefore also, as the circumference BC is to the circum- 
ference EF, so is the angle BGC to the angle EHF, and 
the angle BA C to the angle EDF. 

Therefore etc, 

'■ ■ Q. E. D. 

This proposition as generally given includes a second part relating to sectors 
of circles, corresponding to the following words addt^ to the enunciation : 
" and further the sectors, as constructed at the centres " {hi it not al to/«« art 
[or oiTt] irf)o« rots «rKTpo« ot)hotq'^(voi). There is of course a corresponding 
addition to the "definition" or "particular statement," "and further the sector 
GBOC to the sector HEQF" These additions are clearly due to Theon, as 
may be gathered from his own statenient in his commentary on the ^Sij^tijoJ 
<ru\na^i<i of PtoIemy, " But that sectors in equal circles are to one another as 
the angles on which they stand, has been proved by me in my edition of the 



VI. 33] PROPOSITION 33 -^S 

Elements at the end of the sixth book." Campanus omits them, and P has them 
only in a later hand in the margin or between the lines. Theon's proof scarcely 
needs to be given here in full, as it can easily be supplied. From the equality 
of the arcs BC, CK he infers [in. 29] the equality of the chords BC, CK, 
Hence, the radii being equal, the triangles GBC, GCK are equal in all 
respects [i. 8, 4]. Next, since the arcs BC, CK are equal, so are Xhz arcs 
BAC, CAK. Therefore the angles at the circumference subtended by the 
latter, i.e. the angles in the segments BOC, CPK, are equal [iii. v\\ and the 
segments are therefore similar [in. Def.. iil and equal [in. 24]. 

Adding to the equal segments the equal tnangles GBC^ CCA" respectively, 
we see that 

the sectors GBC, GCK are equal. 

Thus, in equal circles, sectors standing on equal arcs are equal ; and the rest 
of the proof proceeds as in Euclid's proposition. 

As regards Euclid's proposition itself, it will be noted that (i), besides 
quoting the theorem in hi. 27 that in equal circles angles which stand on 
equal arcs are equal, the proof assumes that the angle standing on a greater 
arc is greater and that standing on a less arc is less. This is indeed a suffi- 
ciently obvious deduction from in. 27. 

(2) Any equimultiples whaievtr are taken of the angle BGC and the arc 
BC, and any equimultiples whatever of the angle EHF and the arc EF. 
(Accordingly the words "any (quimuUipUs whaievtr" should have been used in 
the step immediately preceding the inference that the angles are proportional 
to the arcs, where the text merely states that there have been taken of the 
circumference BC and the angle BGC equimultiples BL and BGL.) But, if 
any multipk of an angle is regarded as being itself an angle, it follows that the 
restriction in 1. Deff, 8, 10, ii, 12 of the term angltio an angle less than two 
right angles is implicitly given up ; as De Morgan says, "the angle breaks 
prison," Mr Dodgson {Euclid and his Modern Rivals, p, 193} argues that 
Euclid conceived of the multiple of an angle as so many separate angles not 
added together into one, and that, when it is inferred that, where two such 
multiples of an angle are equal, the arcs subtended are also equal, the argu- 
ment is that the sum total of the first set of angles is equal to the sum total 
of the second set, and hence the second set can be broken up and put 
together again in such amounts as to make a set equal, each to each, to the 
first set, and then the sum total of the arcs will evidently be equal also. If 
on the other hand the multiples of the angles are regarded as single angular 
magnitudes, the equality of the subtending arcs is not inferrible directly from 
Euclid, because his proof of ni. 26 only applies to cases where the angle is 
less than the sum of two right angles, (.'^s a matter of fact, it is a question of 
inferring equality of angles or multiples of angles from equality of arcs, and 
not the converse, so that the reference should have been to in. 27, but this 
does not affect the question at issue.) Of course it is against this view of 
Mr Dodgson that Euclid speaks throughout of " the angle BGL " and " the 
an^e EHN " (ij imo BHA ■yiui'i'a, jj ujro E0N yoivia). I think the probable 
explanation is that here, as in in. 20, 21, 26 and 27, Euclid deliberately took 
no cognisance of the case in which the multiples of the angles in question 
would be greater than two right angles. If his attention had been called to 
the fact that in. 20 takes no account of the case where the segment is less 
than a semicircle, so that the angle in the segment is obtuse, and therefore the 
" angle at the centre " in that case (if the term were still applicable) would be 



»j« BOOK VI [vi. 33 

greater than two right angles, Euclid would no doubt have refused to regard 
the latter as an angle, and would have represented it otherwise, e.g. as the 
sum of two angles or e^ what is left when an angU in the true sense is sub- 
tracted from four right angles. Here then, if Euclid had been asked what 
course he would take if the multiples of the angles in question should be 
greater than two right angles, he would probably have represented them, I 
think, as being equal to so many right anghs plus an angU less than a right 
angle, or so many limes two right angles plus an angle, acute or obtuse. Then 
the equality of the arcs would be the equality of the sums of so many circum- 
ferences, semi-circumferences or quadrants plus arcs less than a semicircle or 
a quadrant. Hence I agree with Mr Uodgson that vi. 33 affords no evidence 
of a recognition by Euclid of " angles " greater than two right angles 

Theon adds to his theorem about sectors the Porism that. As the sector is 
to the sector, so also is the angle to the angle. This corollary was used by 
Zenodorus in his tract -n-fpt uro/ifrpcuv ax^p-ituiv preserved by Theon in his 
commentar)' on Ptolemy's oi/n-afu, unless indeed 'i'heon himself interpolated 
the words {^s S" tom^ut tt/m^ T^tv rofiia, tJ vro E0A yiui'Ja Trp^ t^v viro M@A), 



.1 ,1. 






BOOK VII. 



DEFINITIONS. 



1. An unit is that by virtue of which each of the things 
that exist is called one. • ' 

2. A number is a multitude composed of units. 

3. A number is a part of a number, the less of the 
greater, when it measures the greater ; 

4. but parts when it does not measure it. 

5. Tue greater number is a multiple of the less when 
it is measured by the less. 

6. An even number is that which is divisible into two 
equal parts, 

7. An odd number is that which is not divisible into 
two equal parts, or that which differs by an unit from an 
even number, ■ . ... 

8. An even-times even number is that which is 
measured by an even number according to an even number. 

9. An even-times odd number is that which is 
measured by an even number according to an odd number, 

10. An odd-times odd number is that which is 
measured by an odd number according to an odd number. 



gft "■' BOOK VII [vii. DEFF, II — as 

11. A prime number is that which is measured by an 
unit alone, .. ^ . 

12. Numbers prime to one another are those which 
are measured by an unit alone as a common measure. 

13. A composite number is that which is measured 
by some number. 

14. Numbers composite to one another are those 
which are measured by some number as a common measure. 

15. A number is said to multiply a number when that 
which is multiplied is added to itself as many times as there 
are units in the other, and thus some number is produced. 

16. And, when two numbers having multiplied one 
another make some number, the number so produced is 
called plane, and its sides are the numbers which have 
multiplied one another. 

17. And, when three numbers having multiplied one 
another make some number, the number so produced is 
solid, and its sides are the numbers which have multiplied 
one another. 

18. A square number is equal multiplied by equal, or 
a number which is contained by two equal numbers. 

19. And a cube is equal multiplied by equal and again 
by equal, or a number which is contained by three equal 
numbers. 

20. Numbers are proportional when the first is the 
same multiple, or the same part, or the same parts, of the 
second that the third is of the fourth. 

21. Similar plane and solid numbers are those which 
have their sides proportional. 

22. A perfect number is that which is equal to its own 
parts. 



VII DM. i] DEFINITIONS <- 3>9 



Definition i. 

Movas teric, Kaff 4jv inatrrov tav 6vTtov ty Xryerai. 

lamblichus (fl. ana 300 a.d.) ceKs us (Comm. on Nieemaehus, ed. Pistelli, 
p. 1 1, 5) that the Euclidean definition of an unii or a monad was the definition 
given by " more recent " writers (ol vta/rtpoi.), and that it lacked the words 
"even though it be collective" {ni-v (twmiiijiTuthi %). He also gives (ibid. 
p. 11) a number of other definitions, (t) According to "some of the Pytha- 
goreans," " an unit is the boundary between number and parts " (^omt brrar 
ApSfuiv KQi noftiuai iitSopuir), " because from it, as from a seed and eternal 
root, ratios increase reciprocally on either side," i.e. on one side we have 
multiple ratios continually increasing and on the other (if the unit be sub- 
divided) submultiple ratios with denominators continually increasing. (2) A 
somewhat similar definition is that of Thyniaridas, an ancient Pyth^orean, 
who defined a monad as " limiting quantity " (ir«patVow<ra iroowij!), the 
beginning and the end of a thing being equally an extremity {vipa.%). Perhaps 
the words together with their explanation may hest be expressed by " limit of 
fewness." Theon of Smyrna (p. 18, 6, ed. Hill'er) adds the explanation that 
the monad is " that which, when the multitude ts diminished by way of 
continued subtraction, is deprived of all number and takes an abiding position 
{liBv^) and rest." If, after arriving at an unit in this way, we proceed to divide 
the unit itself into parts, we straightway have multitude again. (3) Some, ac- 
cording to lamblichus (p. r r, 16), defined it as the "form of forms" (dStuK <I8«) 
because it potentially comprehends all forms of number, e.g. it is a polygonal 
number of any number of sides from three upwards, a solid number in all 
forms, and so on, (We are forcibly reminded of the latest theories of number 
as a "Gattung" of "Mengen" or as a "class of classes.") {4) Again an 
unit, says lamblichus, is the first, or smallest, in the category of how many 
(rixrov), the common part or beginning of Aow many. Aristotle defines it as 
" the indivisible in the (category of) quantity," to xari to voaov dSuupfrov 
{Mdaph. 10S9 b 35), ttottoy including in Aristotle continuous as well as 
discrete quantity ; hence it is distinguished from a point by the fact that it 
has not position : "Of the indivisible in the category of, and quA^ quantity, 
that which is every way (indivisible) and destitute of position is called an 
unii, and that which is every way indivisible and has position is a point" 
(Meiaph. io[6b2S). (5) In accordance with the last distinction, Aristotle 
calls the unit " a point without position," ariyiii) afftros {Mtlaph. 1084 b 26), 
(6) Lastly, lamblichus says that the school of Chrysippus defined it in a con- 
fused manner {<niyKr)(V)Uviiii) as " multitude one {ttk^Om tv)," whereas it is 
alone contrasted with multitude. On a comparison of these definitions, it 
would seem that Euclid intended his to be a more popular one than those 
of his predecessors, S);/ui!%c, as Nicomachus called Euclid's definition of an 
iven number. 

The etymological signification of the word floras is supposed by Theon of 
Smyrna (p. 19, 7 — 13) to be either (i) that it remains unaltered if it be 
multiplied by itself any number of times, or (») that it is separated and isolattd 
(^MjaofiHa^at) from the rest of the multitude of numbers. Nicomachus also 
observes (1. 8, a) that, while any number is half the sum (i) of the adjacent 
numbers on each side, (2) of numbers equidistant on each side, the unit is 
mo$i solitary (/iwdtran;) in that it has not a number on each side but only on 
one side, and it is half of the latter done, i.e. of 2. 



aSO BOOK VII [Vll, DEFF, 2—4 



Definition 2. 

The definition of a numbir is again only one out ol many that are on 
record. Nicomachus (i. 7, i) combines several into one, saying that it is 
" a defined multitude (ttXij^o? (ijptcr^«Vor), or a collection of unit.s (^afa£ti>v 
o-uo-ni^ta), or a flow of quantity made Up of units " (ttoitotiitik x^fo *" liova&aiv 
avyKtljufvov). Theon, in words almost identical with those attributed by 
Stobaeus {Edogae, 1. i, 8) to Moderatus, a Pythagorean, says (p. 18, 3—5): 
" A number is a collection of units, or a progression (jrpcHro&trfio'v) of mul- 
titude beginning from an unit and a retrogression (dva7ro8«rfio'«) ceasing at an 
unit." According to lamblichus (p. 10) the description "collection of units" 
(^ovaSujv o-u'crnj/ui) was applied to the how many, i.e. to number, by Thales, 
following the Egyptian view (kq™ to \\.yvimaKov aftioKov), while it was 
Eudoxus the Pythagorean who said that a number was "a defined multitude" 
(n-Xijfio! Kifn.tr \i.ivav). Aristotle has a number of definitions which come to the 
same thing: "limited multitude" (irX^flos- to TrfTrfpao-^tfof, Metapk. lOio a 
13), "multitude" (or "combination") "of units" or "multitude of indivi- 
sibles" (ibid. 1 05 J a 30, 1039 a 12, 1085 b 22), "several outi" i^va. TrXtim, 
Phys. HI. 7, 207 b 7), "multitude measurable by one" {Meiaph. 1057 a 3) 
and " multitude measured and multitude of measures," the " measure " being 
unity, TO tv {ibid. 1088 a 5). 

Definition 3. 

By a pari Euclid means a submultiple, as he does in v. Def. i, with which 
definition this one is identical except for the substitution of number (aptSyaojJ 
for magnitude (/tiyfffo^) ; cf note on v. Def. 1 , Nicomachus uses the word 
"submultiple" (vwmroXKaTrXairwi) also. He defines it in a way corre.'jponding 
to his definition of multiple (see note on Def 5 below) as follows (1, 18, 2): 
" The submultiple, which is by nature first in the division of inequality 
(called) less, is the number which, when compared with a greater, can 
measure it more times than once so as to fill it exactly (jrAj^poufru^)." Simi- 
larly sub-double (iIiroStTrXao-io;) is found in Nicomachus meaning half, and 
so on. 

Definition 4. 

Mept) S{, orav /i^ KaTO^tTp^. 

By the expression parts {fiipij, the plural of ftipirt) Euclid denotes what we 
should call A proper fraition. That is, a pari being a submultiple, the rather 
inconvenient term parts means any numbtrr of such submultiples making up 
a fraction less than unity. I have not, found the word used in this special 
sense elsewhere, e.g. in Nicomachus, Theon of Srnyrna or lamblichus, except 
in one place of Theon (p. 79, 26) where it is used of a proper fraction, of 
which 1^ is an illustration. 



viu OBFF. s— 8] NOTES ON DEFINITIONS 2—8 381 

Definition $. 

The definition of a multiple is identical with that in v. Def, 2, except that 
the masculine of the adjectives is used agreeing with npi^pi's understood 
instead of the neuter agreeing with fitytOo^ understood. Nicomachus (i. 18, 
i) defines a multiple as being "a species of the greater which is naturally- 
first in order and origin, being the number which, when considered in com- 
parison with another, contains it in itself completely more than once." 

Definitions 6, 7, 

6. ApniK apt6)iot ivTiv 6 Sijfa Siaipovittro<,. 

7. Tlfpicro'os oc fLrj oiatfjo^fi.H'Oi ^t)^a. ^ [o] ftovd^i Sia^pbrv Aprlov &pi$fLOv. 

Nicomachus (1. 7, a) somewhat amplifies these definitions of eve/t and oild 
numbers thus, "That is evtri which is capable of being divided into two 
equal parts without an unit falling in the middle, and that is odd which cannot 
be divided into two equal parts because of the aforesaid intervention (/i«ri- 
niay) of the unit." He adds that this definition is derived " from the popular 
conception " {Ik tij! Sij/iuBout uToAij^fon). In contrast to this, he gives (t. 7, 3) 
the Pythagorean definition, which is, as usual, interesting. "An av» number 
is that which admits of being divided, by one and the same operation, into the 
greatest and the least (parts), greatest in size (infKtKo-njTi) hut least in quantity 
(irmroDTTt). ..while an odd number is that which cannot be so treated, but is 
divided into two unequal parts." That is, as lamblichus says (p. 12, 2—9), an 
even number is divided into parts which are the p-e<iUst possible "parts," namely 
halves, and into the ftwtst possible, namely two, two being the first " num- 
ber" or "collection of units." According to another ancient definition quoted 
by Nicomachus (i. 7, 4), an even number is that which can be divided both 
into two equal parts and into two unequal parts {except the first one, the 
number 2, which is only susceptible of division into eiiuals), but, however it 
is divided, must have its two parts of the same kind, i.e. both even or both 
odd ; while an odd number is that which can only l)e divided into two 
unequal parts, and those parts always of differait kinds, i.e. one odd and 
one even. Lastly, the definition of odd and even "by means of each other" 
says that an odd number is that which differs by an unit from an even 
number on both sides of it, and an even number that which differs by an 
unit from an odd number on each side. This alternative definition of an 
odd number is the same thing as the second half of Euclid's definition, " the 
number which differs by an unit from an even number." This evidently 
pre-Euclidean definition is condemned by Aristotle as unscientific, because 
odd and even are coordinate, both being differentiae of number, so that one 
should not be defined by means of the other {Topics vi. 4, 142 b 7 — 10). 

Definition 8. 

^Kpnaxi^ aprio^ aptVfjLO^ temv o vtto ofyrlov dptBfAOv fitrpovfitvo^ Kara apTiOV 
iipi$fi6v. 

Euclid's definition of an ei'en-times even number dift'ers from that given by 
the later writers, Nicomachus, Theon of Smyrna and lamblichus ; and the 
inconvenience of it is shown when we come to ix. 34, where tt is proved 



iSi BOOK Vn '• [vit. DCFF. 8, 9 

ihat A. certain sort of number is Aa/A "even-times even "and "even-times odd." 
According to the more precise classification of the three other authorities, the 
" even-times even " and the " even-times odd " are mutually exclusive and are 
two of three subdivisions into which even numbers fall. Of these three sub- 
divisions the "even-times evt;ii " and the "tven-timcs odd" form the extremes, 
and the "odd-times even" is as it were intermediate, showing the character 
of both extremes (cf, note on the following definition). The even-times emn is 
then the number which has its halves even, the halves of the halves even, and 
so on, until unity is reached. In short the evtn-times evin number is always 
of the form z". Hence lamblichus (pp. lo, zi) says Euclid's definition of it 
as that which is measured by an even number an even number of times is 
erroneous. In support of this he quotes the numljer 24 which is four times 6, 
or six times 4, but yet is not " even times even " according to Euclid himself 
(oiSi na/ auToi'), by which he musjl apparently mean that 24 is also 8 times 3, 
which does not satisfy Euclid's definition, '['here can however be no doubt that 
Euclid meant what he said in his definition as wt have it ; otherwise ix. 32, 
which proves that a number of the form 2" is even-limes even only, would be quite 
superfluous and a mere repetition of the definition, while, as already stated, 
IX. 34 clearly indicates Euclid's view- that a numlier might at the same time 
be both even. times even and even-times odd. Hence the ftdt'ias which some 
editor of the commentary of I'hiloponus on Nicomachus found in some 
copies, making the definition say that the even-times even number is only 
measured by even numbers an even number of times, is evidently an interpo- 
lation by some one who wished to reconcile Euclid's definition with the 
Pythagorean (cf Heiberg, Eiiklid-siudien, p. a 00). 

A consequential characteristic of the series of even-times even numbers 
noted by Nicomachus brings in a curious use of the word Suvq^i (generally 
power in the sense of square, or square root). He says (). 8, 6 — 7) that any 
part, i.e. any submultiple, of an even -times even number is called by an even- 
times even designation, while it also has an even-times even value {it is 
apTtciitts dpTmSurafioi'} when expressed as so many actual units. That is, the 

-,th part of 2" (where m is less than «) is called after the even-times even 

number z™, while its actual value (Sufa/ut) in units is 2"-"*, which is also an 
even-times even number. Thus all the parts, or submultiples, of even-times 
even numbers, as well as the even-times even numbers themselves, are con- 
nected with one kind of number only, the even. 

Definition 9. 

'ApTiaKit S{ TTtpunrd; hrtw h viro iptiov ifuB^oxt fLcrpovjucvos Kara npurvvi' 

Euclid uses the term even- times odd (dpTniic« vipvaao^), whereas Nicomachus 
and the others make it one word, even-odd (aprtairifiirro^). According to the 
stricter definition given by the latter (i, 9, i), the even-odd number is related to 
the even-limes even as the other extreme. It is such a number as, when once 
halved, leaves as quotient an odd number; that is, it is of the form j(2«+ i). 
Nicomachus sets the even-odd numbers out as follows, 

6, 10, 14, 18, zz, 26, 30, etc. 
In this case, as Nicomachus observes, any part, or submultipie, is called by a 
name not corresponding in kind to its actual value (Sura/iw) in units. Thus, 



viL DEF. 9] NOTES ON DEFINITIONS 8, 9 183 

in the case of i3, the ^ part is calied a.rter the even number 2, but its va/ue is 
the odd number 9, and the Jrd part is called after the odd number 3, while its 
value is the even number 6, and so on. 

The third class of even numbers according to the strict subdivision is the 
odd-even {TrtpunrdpTioi). Numbers are of this class when they can be halved 
twice or more times successively, but the quotient left when they can no 
longer be halved is an odd number and not unity. They are therefore of 
the form 2"'^'{7« + 1), where », nt are integers. They are, so to say, inter- 
mediate between, or a mixture of, the extreme classes eiien-times tvin and even- 
odd, for the following reasons, (r) Their subdivision by 3 proceeds for some 
way like that of the even-times even, but ends in the way that the division of 
the even-odd by i ends. (2) The numbers after which submultiples are 
called and their value {Sv'raj««) in units may be both of one kind, i.e. both odd 
or both even {as in the case of the even-times even), or again may be one odd 
and one even as in the case of the even-odd. For example »4 is an odd-even 
number; the ^ ih, tV'^'i ir'h or ^ parts of it are even, but the Jrd part of it, 
or 8, is even, and the Jth part of it, or 3, is odd. (3) "Nicomachus shows 
(i. 10, 6 — 9) how to form all the numbers of the odd-even class. Set out two 
lines (a) of odd numbers beginning with 3, {fi) of even-times even numbers 
beginning with 4, thus : 

(a) 3. S> 7i 9i r»> »3. >S et_c. 

{b) 4, 8, 16, 3*, 64, 128, 156 etc. 

Now multiply each of the first numbers into each df the second row. Let 
the products of one of the first into all the second set make horizontal rows ; 
we then get the rows 

12,24, 48, 96,192, 384, 768 etc. 

20, 40, 80, 160, 320, 640, 1280 etc, 

28, 56, 1 1 J, i24, 448, 896, 179J etc. ,•- 

36, 72, 144, 288, 576, 1152, 2304 etc, 
and so on. 

Now, says Nicomachus, you will be surprised to see (^(Tfcrtrat trm ftiupMr- 
rwi) that (a) the vertical rows have the property of the ezfen-odd series, 6, 10, 
14, 18, 22 etc., viz, that, if an odd number of successive numbers be taken, 
the middle number is half the sum of the extremes, and if an even number, 
the two middle numbers together are equal to the sum of the extremes, 
(t) the horizontal rows have the property of the even-times even series 4, 8, 16 
etc., viz. that the product of the extremes of any number of successive terms 
is equal, if their number be odd, to the square of the middle term, or, if their 
number be even, to the product of the two middle terms. 

Let us now return to Euclid. His 9th definition states that an even-timef 
odd number is a number which, when divided by an even number, gives an 
odd number as quotient. Following this definition in our text comes a loth 
definition which defines an odd-times even number ; this is stated to be a 
number which, when divided by an odd number, gives an even number as 
quotient. According to these definitions any even-times odd number would 
also be odd-times even, anti, from the fact that lamblichus notes this, we may 
fairly conclude that he found Def, 10 as well as Def. 9 in the text of Euclid 
which he used. But, if both definitions are genuine, the erjunciations of ix. 33 
and IX. 34 as we have them present difficulties, ix. 33 says that " If a num- 
ber have its half odd, it is even-times odd only " ; but, on the assumption that 



^ . ,; BOOK VII -,!• [vir. DKKf. 9— 11 

both definitions are genuine, Ihis would not be true, for the number would be 
odd-timts even as well. ix. 34 says that " If a number neither be one of those 
which are continually doubled from 2, nor have its half odd, it is both even- 
times even and even- times odd." The term odd-timts even {irtpuraaKK aprtot) 
not occurring in these propositions, nor anywhere else after the definition, that 
definition liecomes superfluous. Iambi ichus however {p. 24, 7 — m) quotes 
these enunciations differently. In the first he has instead of " even-times odd 
only " the words " both tven-timei odd and odd-times even " ; and, in the second, 
for " both even-times even and even-times odd " he has " is both even-times 
even and at the same time even-times odd and odd-times even." In both 
cases therefore " odd-times even " is added to the enunciation as lamblichus 
had it^ the words catrnot have been added by lamblichus himself because 
he himself does not use the term odd-timts even, but the one word odd-even 
(rtfiuTirafyno^). In Order to get over the difficulties involved by Def, 10 and 
these differences of reading we have practically to choose between (i) accept- 
ing lamblichus' reading in all three places and (2) adhering to the reading of 
our Mss. in ix. 33, 34 and rejecting Def. 10 altogether as an interpolation. 
Now the readings of our text of ix. 33, 34 are those of the Vatican MS. 
and the Theonine mss. as well ; hence they must go back to a time before 
Theon, and must therefore be almost as old as those of lamblichus. 
Heiberg considers it improbable that Euclid would wish to maintain a point- 
less distinction between even-times odd and odd-times even, and on the whole 
concludes that IJef. [O was first interpolated by some ignorant person who 
did not notice the difference between the Euclidean and Pythagorean clissi- 
fication, but merely noticed the absence of a definition of odd-times even 
and fabricated one as a companion to the other. When this was done, it 
would be easy to see that the statement in )X. 33 that the number referred 
to is " even-times odd only " was not strictly true, and that the addition of 
the words "and odd-times even" was necessary in ix. 33 and tx. 34 as 
well. 

Definition 10, 

n*p«r<raKW Be letpuruoi apSjtan iarw viro irtpurami dpiff/iov fierpois/it™? 
Kara Trtpt(r<rov dpiOfxov* 

The Olid-times add number is not defined as such by Nicomachus and 
lamblichus ; for them these numbers would apparently belong to the «»«- 
^osite subdivision of odd numbers. Theon of Smyrna on the other hand 
says (p. 23, 21) that odd-times odd was one of the names applied to prime 
numbers (excluding 2), for these have two odd factors, namely i and the 
number itself. This is certainly a curious use of the term. 

Definition ii. 

IXpitfTiK d,pSiia% ttTTiv b /tovaSi /lovij ftfrpttijitvov. 

A prijne number {vfiS/ro^ iptSjtios) is called by Nicomachus, Theon, and 
lamblichus a " prime irnrf inwm/oj(V^ (iovk^trtre) number." Theon (p. 23, 9) 
defines it practically as Euclid does, viz. as a number "measured by no number, 
but by an unit only." Aristotle too says that a prime number is not measured by 
any number (Ana/, post. ti. 13, 96 a 36), an unit not being a number [Metaph, 
1088 a 5), but only the beginning of number (Theon of Smyrna says the same 
thii^, p. t\, 13). According to Nicomachus (1. 11, a) the prime number is a 



VII. DEFK. [I, 12] NOTES ON DEFINITIONS 9—12 285 

subdivision, not of numbers, but of odd numbers; it is "an odd number 
which admits of no other part except that which is called after its own name 
(n-apiiJniftov <avri2)." The prime numbers art; 3, 5, 7 etc., and (here is no 
submuUiple of 3 except ^rd, no subtnultipiL' of 1 1 except y j th, and so on. hi 
all these cases the only submultiple is an unit. According to Nicomachus 3 
is the first prime number, whereas Aristotle {Topics viii. 2, 157 a 39) regards 
a as a prime number ; "as the dyad is the only even number which is prime," 
showing that this divergence from the l^ythagortan doctrine was earlier than 
Euclid. The number 2 also satisfies Euchd's definition of a prime number, 
lamblichus (p. 30, 27 sqq.) makes this the ground of another attack upon Euclid. 
His argument (the text of which, however, leavijs much to be desired) appears 
to be that i is the only even number which has no other part except an 
unit, while the subdivisions of the even, as previously explained by him (the 
ti-en-timts even, the even-odd, and odd-even), all exclude primeness, and he has 
previously explained that 2 is pottniially even^xld, being obtained by 
multiplying by 2 i\\^ potentiaiiy odd, i.e. the unit; hence 2 is regarded by him 
as bound up with the subdivisions of even, which exclude primeness. 'I'heon 
seems to hold the same view as regards i, but supports it by an ap|)arent 
circle. A prime number, he says (p. 23, 14 — 23), is also called odd-times odd; 
therefore only odd numbers are prime and in composite. Even numbers are 
not measured by the unit alone, except i, wliich therefore (p. 24, 7) is odd-Aiti' 
{itiparirati&rfi) without being prime. 

A variety of other names were applied to prime numbers. We have 
already noted the curious designation of them as add-titnts odd. According to 
lamblichus (p. 27, 3 — 5) some called them evthymtiric ((uSu^fTpwds), and 
Thymaridas rectilinear (<iSv7pa/tfuitOT), the ground being that they can only be 
set out in one dimension with no breadth (iirXar^s yap iv i-g Mtati iiji' iv 
liovot iiuTTafitun). The same aspect of a prime number is also expressed by 
Aristotle, who (Metaph. 1020 b 3) contrasts the composite number with that 
which is only in one dimension (fioi'oi' i<^ tv cuv). Theon of Smyrna (p. 23, 1 2) 
gives ■)>pa^/iutdt {linear) as the alternative name instead of cv^uypo^oidt. In 
either ease, to make the word a proper description of a prime nun>ber we have 
to understand the word only ; a prime number is that which is linear, or 
rectilinear, only. For Nicomachus, who uses the form linear, expressly says 
(11. 13, 6) that all numbers are so, i.e. all can be represented as linear by dots 
to the required amount placed in a line. 

A prime number was called prime or first, according to Nicomachus 
(1. II, 3), because it can only be arrived at by putting together a certain 
number of units, and the unit is the beginning of number (cf. Aristotle's 
second sense of irpiuTos "as not being composed of numbers" wi /jltj o-uymurSat 
i( dpiSfiMv, Anal. Post. \\. 13, 96 a 37), and also, according to lamblichus, 
because there is no number before it, being a collection of units (^ofaSur 
<riim]yut), of which it is a multiple, and it appears firsl as a basis for other 
numbers to be multiples of. 



Definition 12. 

EtpioT'Ol Jrpot a\AijXou! apSpai vaw oi /iom& judv]) fttTpoi,! - ■jt koh-iJ liirpi^. 

By way of further emphasising the distinction between ''prime" and 
"prime to one another," Theon of Smyrna (p. 23, 6—8) calls the former 
" prime aholtttefy " (dwKm), and the latter " prime to one another and not 



016 - . BOOK Vn ■ .. [V!I. DEFP. I!!— 14 

absolttfdy" or *^noi in themseives" (oi xaff aarmi). The latter (p. 44, 3 — lo) 
are " measured by the unit [sc. only] as common measure, even though, taken 
by themselves {w jrpo? Javrm), they be measured by some other numbere." 
From Theon's illustrations it is clear that with him as with Euclid 
a. number prime to another may be even as well as odd. In Nicomachus 
(i. 1 1, i) and lambllchus (p. id, 19), on the other hand, the number which is 
" in itself secondary (Scvre^Mt) and composite (tr!ni6tTo%), but in relation to 
another prime and incomposite," is a subdivision of odd. I shall call more 
particular attention to this difference of classification when we have reached 
the definitions of " composite " and " composite to one another " ; for the 
present it is to be noted that Nicomachus (1. 13, i) defines a number prime to 
another after the same manner as the absolutely prime ; it is a number which 
" is measured not only by the unit as the common measure but also by some 
other measure, and for this reason can also admit of a part or parts called by 
a difTerent name besides that called hy the same name (as itself), but, when 
examined in comparison with another number of similar character, is found 
not to be capable of being measured by a common measure in relation to the 
other, nor to have the same part, called by the same name as (any of) those 
simply (air\c«) contained in the other; e.g. 9 in relation to 25, for each of 
these is in itself secondary and composite, but, in comparison with one 
another, they have an unit alone as a common measure and no part is called 
by the same name in both, but the third in one is not in the other, nor is the 
fifth in the other found in the first." 

Definition 13. . . 

SwrftTOt optC/«k ^<rr(v o afiSfu^ tiki /iMrpoufitvot. 

Euclid's definition of compositt is again the same as Theon's definition 
of numbers "composite in relation to themselves," which (p. 24, 16) are 
" numbers measured by any less number," the unit being, as usual, not 
regarded as a number. Theon proceeds to say that " of composite numbers 
they call those which are contained by two numbers plane, as being 
investigated in two dimensions and, as it were, contained by a length and a 
breadth, while (they call) those (which are contained) by three (numbers) 
iolid, as having the third dimension added to them," To a similar effect is 
the remark of Aristotle {Mtlaph. loio b 3) that certain numbers are 
" composite and are not only in one dimension but such as the plane and the 
solid (figure) are representations of (^i^iijjiia), these numbers being so many 
times so many (irocraKK iroa-oi), or so many times so many times so many 
(iroo-am! wtwaitt* voaai) respectively." These subdivisions of composite 
numbers are, of course, the subject of Euclid's definitions 17, 18 respectively. 
Euclid's composite numbers may be either even or odd, like those of Theon, 
who gives 6 as an instance, 6 being measured by both » and 3. 

Definition 14. 

liirpit. 

Theon (p. 44, 18), like Euclid, defines numbers eomposite to one another as 
*' those which are measured by any common measure whatever " (excluding 
unity, as usual). Theon instances 8 and ti, with i as common measure, and 
6 and 9, with 3 as common measure. 



vir. DErF, 14— :6] NOTES ON DEFINITIONS Ji— 16 sS? 

As hinted above, there is a great difference between Euclid's classification 
of prime and composite numbers, and of numbers prime and comp>osite 
to one another, and the classification found in Nicomachus (1. 11 — 13) and 
lamblichus. According to the latter, all these kinds of numbers are sub- 
divisions of the class of odd numbers only. As the class of even numbers is 
divided into three kinds, (i) the even-times even, (2) the even-odd, which 
form the extremes, and (3) the odd-even, which is, as it were, intermediate to 
the other two, so the class of odd numbers is divided into thtee, of which the 
third is again a mean between two extremes. The three are : 

(i) t\\t primt and ineomposite, which is like Euclid's prime number except 
that it excludes 2 ; 

{i) the ieconiary and composite, which is "odd because it is a distinct 
part of one and the same genus {Sia tu ii jvot kcu tov avroii yirous Stciit<Kpur0(u) 
but has in it nothing of the nature of a first principle (ap;(«iSft) ; for it arises 
from adding some other number (to itself), so that, besides having a part 
called by the same name as itself, it possesses a part or parts called by another 
name." Nicomachus cites 9, 15, 21, 25, 27, 33, 35, 39. It is made clear that 
not only must the factors be both odd, but they must all be prime numbers. 
This is obviously a very inconvenient restriction of the use of the word 
composite, a word of general signification. 

(3) is that which is "secondary and composite in itself but prime and 
ineomposite to another" The actual words in which this is defined have been 
given above in the note on Def. 12. Here again all the factors must be odd 
and prime. 

Besides the inconvenience of restricting the term composite to odd numbers 
which are composite, there is in this classification the further serious defect, 
pointed out by Nesselmann {Die Algebra der Griechen, 1842, p. 194), that 
subdivisions (2} and (3) overlap, subdivision (2) including the whole of 
subdivision {3). The origin of this confusion is no doubt to be found in 
Nicomachus' perverse anxiety to be symmetrical ; by hook or by crtxik he 
must divide odd numbers into three kinds as he had divided the even. 
lamblichus (p. 28, 13) carries his desire to be Ic^cal so far as to point out 
why there cannot be a fourth kind of number contrary in character to (3), 
r\amely a number which should be "prime and ineomposite in itself, but 
secondary and composite to another " ! 



Definition 15. 

'ApiB/iAi dpSfthv iroXAaTrAxuruL^cif Xiynat, oraf, 3crot »uriv h earr^ )tov^K, 
T«ravrax(! inivTi$^ o iraXXaiTAairiafD^ci'Ov, ««! yiyijrrai tk. 

This is the well known primary definition of multiplication as an 
abbreviation of addition. 

Definition 16. 

iviirtoo^ KaktiTat^ irAcvpat Bi airroZ ol vo\kair\aeTaurayT€^ dXXijKom dpiBfioL 

The words plane and solid applied to numbers are of course adapted from 
Iheir use with reference to geometrical figures. A number is therefore called 
linear (ypaiifuKVi) when it is regarded as in one dimension, as being a lengtli 



a«8 BOOK vn [vii. def. i6 

(lAiJKiys), When it takes another dimension in addition, namely breadth 
(TrAaros), it is in two dimensions and becomes plane (in-iVtScn). The 
distinction bet wet! n a plane and a plane number is marked by the use of the 
neuter in the former case, and the masculine, agreeing with aptS/io!, in the 
latter case. So witli a square and a square number, and so on. Tlie most 
obvious form of a plane number is clearly that corresponding to a rectangle in 
geometry ; the number is the product of two linear numbers regarded as sides 
{TrKfvpai) forming the length and breadth respectively. Such a number is, as 
Aristotle says, "so many times so many," and a plane is its counterpart 
{/ii/iTj/ia). So I'lato, in the Thcaeieiiis (147 E — 148 b), says : "We divided all 
numbers into two kinds, ( i ) that which can be expressed as equal multiplied 
by equal (tov ivvi.\i.(vav Xaov Xaixtx yiyno-Sai), and which, likening its form to 
the square, we called square and equilateral ; {«) that which is intermediate, 
and includes 3 and 5 and every number which cannot be expressed as equal 
multiplied by e<|uat, but is cither less times more or more times less, being 
always "contained by a greater and a less side, which number we likened to 
the oblong figure (ttpo/ijjkh axnt^'") and calleo an obhng number.... Such 
lints therefore as square tlie equilateral and plane nujnber fi.e, which can 
form a plane number with equal sides, or a square] we defined as length 
iltijxo'i) ; but such as square the oblong (here fTfpo/iijKJ)^) [i.e, the square of 
which is equal to the oblong] we called roots {^vulitai) as not being com- 
mensurable with the others in length, but only in the plane areas (^irwiBow), 
to which the squares on them are equal {a Suyavrai)." This passage seems 
to make it clear that Plato would have represented numbers as Euclid does, 
by straight lines proportional in length to the numbers they represent (so far 
as practicable) ; for, since 3 and S are with Plato oblong numbers, and lines 
with him represent the sides of oblong numbers (since a line represents the 
" root," the square on which is equal to the oblong), it follows that the unit 
representing the smaller side must have been represented as a line, and 3, the 
larger side, as a line of three times the length. But there is another possible way 
of representing numbers, not by lines of a certain length, but hypain/s disposed 
in various ways, in straight lines or otherwise. lamblichus tells us (p. 56, 27) 
that " in old days they represented the ijuantuplicities of number in a more 
natural way (^ucriKuir (;»>') by splitting them up into units, and not, as in our 
day, by symbols" (o-«ftj3oXuiiu?). Aristotle too (Metaph. 1092 b 10) mentions 
one Eurytus as having settled what number belonged to what, such a number 
to a man, such a number to a horse, and so on, "copying their shapes" 
(reading rovruii', with Zeller) ^' with pebbles {rms ^^mi), just as those da who 
arrange numbers in the forms of triangles or squares." We accordingly find 
numbers represented in Nicomachus and Theon of Smyrna by a number of 
a's ranged like points according to geometrical figures. According to this 
system, any number could be represented by points in a straight line, in which 
case, says lamblichus {p. 56, 26), we shall call it rectilinear because it is 
without breadth and only advances in length (oirXaTttfi inl isavav to ^7k« 
vpotunv). The prime number was called by Thymaridas rectilinear par 
excellence, because it was without breadth and in one dimension only (iift fc 
^vov SiuTTOfitvin). By this must hi meant the impossibility of representing, 
say, 3 as a plane number, in Plato's sense, i.e. as a product of two numbers 
corresponding to a rectangle in geometry ; and this view would appear to rest 
simply upon the representation of a number by points, as (iistinct from lines. 
Three dots in a straight line would have no breadth ; and if breadth were 
introduced in the sense of producing a rectangle, i.e. by placing the same 



VII. DEF. i6] NOTE ON DEFINITION 16 "^ 

number of dots in a second line below the first line, the first f/ane number 
would be 4, and 3 would not be a plane number at ali, as Plato says it is. It 
seems therefore to have been the alternative representation of a number by 
points, and not lines, which gave rise to the different view of a plane number 
which we find tn Nicomachus and the rest. By means of separate points we 
can represent numbers in geometrical forms other than rectangles and squares. 
One dot with two others symmetrically arranged below it shows a triangle, 
which is a figure in two dimensions as much as a rectangle or parallelogram is. 
Similarly we can arrange certain numbers in the form of regular ptnlagons or 
other polygons. According therefore to this mode of representation, 3 is the 
first plane number, being a triangular number. The method of formation of 
triangular, square, pentagonal and other polygonal numbers is minutely 
described in Nicomachus (11. 8 — 11), who distinguishes the separate series of 
gnomons belonging to each, i.e. gives the law determining the number which 
has to be added to a polygonal number with n in a side, in order to make it 
into a number of the same form but with n + i in a side (the addend being of 
course the gnomon). Thus the gnomon ic series for triangular numbers is 
'> 2i 3) At 5"- 't that for squares i, 3, 5, 7... ; that for pentagonal numbers 
I, 4, 7, 10,,., and so on. The subject need not detain us longer here, as we 
ate at present only concerned with the different views of what constitutes a 
plane number. 

Of plane numbers in the Platonic and Euclidean sense we have seen that 
Plato recognises two kinds, the square and the oblong (vpoii-^icf}<s or htpoinjieifs). 
Here again Euclid's successors, at all events, subdivided the class more 
elaborately. Nicomachus, Theon of Smyrna, and lamblichus divide plane 
numbers with unequal sides into (i) Irtpofi-iJKtK, the nearest thing to squares, 
viz. numbers in which the greater side exceeds the less side by i only, or 
numbers of the form n(n+ i), e.g. i . J, a - 3, 3 . 4, etc. {according to Nico- 
machus), and (2) wpo{tT^Kfi<s, or those whose sides differ by z or more, i.e. are of 
the form n(n + m), where m is not less than z (Nicomachus illustrates by 2 . 4, 
3 . 6, etc.). Theon of Smyrna (p. 30, 8 — 14) makes wpofujuii! include Irf/jo/iijMtt, 
saying that their sides may differ by i or more; he also speaks of parallelogram- 
numbers as those which have one side different from the other by 3 or more ; 
I do not find this latter term in Nicomachus or lamblichus, and indeed it 
seems sufterfluous, as parallelogram is here only another name for oblong, 
lamblichus (p. 74, z 3 sqq,), always critical of Euclid, attacks him again here 
for confusing the subject by supposing that the htpofi^inp number is the pro- 
duct of any two different numbers multiplied together, and by not distinguishing 
the oblong (irpo^ifiti;;) from it : " for his definition declares the same number 
to be square and also htpoit^xtft, as for example 36, 16 and many others : 
which would be equivalent to the odd number being the same thing as the 
even." No importance need be attached to this exaggerated statement ; it is 
in any case merely a matter of words, and it is curious that Euclid does not in 
feet use the word fr«po/iijir7t of numiers at all, but only of geometrical oblong 
figures as opposed to squares, so that lamblichus can apparendy only have 
inferred that he used it in an unorthodox manner from the geometrical use of 
the term in the definitions of Book i. and from (he fact that he does not give 
the two subdivisions of plane numbers which are not square, but seems only 
to divide plane numiers into square and not-square. The aigument that 
Ircpo^ijKttf numbers are a natural^ and therefore essential, subdivision 
lamblichus appears to fotmd on the method of successive addition by which 
they can be evolved ; as square numbers are obtained by successively adding 



i^o •'-■ BOOK VII [vii. DEFF. i6, 17 

odd numbers as gnomons, so ir<pci/ij)Kf« are obtained by adding even numbers 
as gnomons. Thus i.z = 2, 2.3 = 2 + 4, 3,4-2 + 4 + 6, and so on. 



Definition 17. 

{TTfpm i(mv, v\tvpal Sc airoO oi jroAAairXacricio-aiTf! dAAi^Aou; apSfioi. 

What has been said of the two apparently different ways of regarding a 
plane number seerns to apply equally, mutatis mutandis, to the definitions of a 
solid number. Aristotle regards it as a number which is so many times so 
many times so many (jrpcrojtu irt«rai«! itoaoi). Plato finishes the passage about 
lines which represent the sides of square numbers and lines which are roots 
i^vva/itK), i.e. the squares on which are equal to the rectangle representing a 
number which is oblong and not square, by adding the words, " And another 
similar property belongs to solids " (kqI ircpi to o-rtpta aAAo toioZtov). That is, 
apparently, there would be a corresponding term to root (ftJca/in) — practically 
representing a surd— to denote the side of a cube equal to a parallelepiped 
representing a solid number which is the product of three factors but 
not a cube. Such is a solid number when numbers are represented by 
straight lines : it corresponds in general to a parallelepiped and, when all 
the factors are equal, to a cube. 

But again, if numbers be represented by points, we may have solid numbers 
(i.e. numbers in three dimensions) in the form of pyramids as well. The first 
number of this kind is 4, since we may have three points fonTiing an 
equilateral triangle in one plane and a fourth point placed in another plane. 
The length of the sides can be increased by i successively ; and we can have 
a series of pyramidal numbers, with triangles, squares or polygons as bases, 
made up of layers of triangles, squares or similar polygons respectively, each 
of which layers has one less in the side than the layer below it, until the top 
of the pyramid is reached, which of course, is one point representing unity, 
Nicomachus (11, 13 — 16), Theon of Smyrna {p. 41 — 2), and lamblichus 
(P- 9S> '5 sqq.), all give the different kinds c>{ pyramidal solid numbers in 
addition to the other kinds. 

These three writers make the following further distinctions between solid 
numbers which are the product of three factors. 

1. First there is the equal by equal by equal (Icrant to-aittt urof), which is, 
of course, the cube. 

2. The other extreme is the unequal by unequal by unequal {aVto-ciKis 
o.vuia.Ki.% ai'io-ot), or that in which all the dimensions are different, e.g. the 
product of 2, 3, 4 or 2, 4, 8 or 3, 5, la. These were, according to Nicomachus 
(11, 16), called scalene, while some called them a^tjvlanoi (wedge-shaped), others 
tr^i/KUTKot (from tr^ij'f, a wasp), and others ^mitlantot {altar-shaped). Theon 
appears to use the last term only, while lamblichus of course gives all three 
names. 

3. Intermediate to these, as it were, come the numbers " whose planes 
form frepo/nJ«(i numbers" (i.e. numbers of the form*«(« + i)). These, says 
Nicomachus, are QaXXei parallelepipedal. 

Lastly come two classes of such numbers each of which has two equal 
dimensions but not more. 



VII. DEPP. 17— 19I NOTES ON DEFINITIONS id— 19 i0 

4. If the third dimension is less than the others, the number b efual ly 
tquai iy less (uraKtt itroi i'Aairoi'aicw) and is called a plinth (nrXicSw), e.g. 
8.8.3. 

5- If the third dinnensioti is greatei than the others, the number is equal 
by equal by greater (ktb'ms ujik fniioraitw) and is called a beam (&okk), e.g, 
3.3.7. Another name for this latter kind of number (according to 
lambltchus) was ernjXU (diminutive of onjXij). 

Lastly, in connexion with pyramidal numbers, Nifcomachus (11. 14, 5) dis- 
tinguishes numbers corresponding X^i frusta of pyramids. These are truncated 
(mi^upoi), twUe-truntated (BtKoXoupot), thrict-fruncated (rpiKoAoupm) pyramids, 
and so on, the term being used mostly in theoretic treatises («v mr/ypdiiftairi 
fiakuTTa To« StiofnifumKoU). The truncated pyramid was formed by cutting 
off the point forming the vertex. The twice-truncaied was that which lacked 
the vertex and the next plane, and so on. Theon of Smyrna (p. 42, 4) only 
mentions the truncated pyramid as " that with its vertex cut off" (ij ttjc 
Kofsii^ijv diroTiTixsjiLivrj), saying that some also called it a trapezium, after the 
similitude of a plane trapezium formed by cutting the top off a triangle 
by a straight line parallel to the base. 



Definition 18. 

Tcrpaywvo; &.pSlx6% i<rr\y h (cranf uro( ^ [6j vtro Svo wrtav &pi$ftiZv ntpi- 

A particular kind of square distinguished by Nicomachus and the rest was 
the square number which ended (in the decimal notattor) with the same 
number as its side, e.g. i, 25, 36, which are the squares of r, 5 and 6. These 
square numbers were called cyclic (kukXmw) on the analogy of circles in 
geometry which return again to the point from which they started. 



Definition 19, 

KiJjSps Si 6 ttraKit ta-ov liraicK ^ [i] inro rpiuv t(Fii>r Jpifl/uuv wipiixontum. 

Similarly cube numbers which ended with the same number as their sides, 
and the squares of thosr sides aisa, were called spherical (o-f^tpiKof) or reatrrtnt 
{a.iraiaaassta.fma\). One might have expected that the term spherical would be 
applicable also to the cubes of numbers which ended with the same digit as the 
side but not necessarily with the same digit as the square of the side also. 
E.g, the cube of 4, i.e. 64, ends with the same digit as 4, but not with the 
same digit as 1 6, But apparently 64 was not called a spherical number, the 
only instances given by Nicomachus and the rest being those cubed from 
numbers ending with 5 or 6, which end with the same digit if squared. A 
spherical number is in fact derived from a circular number only, and that by 
adding another equal dimension. Obviously, as Nesselmann says, the names 
cyclic and spherical applied to numbers appeal to an entirely different principle 
from that on which the figured numbers so far dealt with were formed. 



jtga BOOK VII 1 1 /• . [vii. DEF. ao 



Definition 2a 

£(7ttic($ ]j iroXAairXdmof i^ to awo fiipo^ ^ ra aura ftc^ ONrtv, 

Euclid does not give in this Book any definition of latio, doubtless because 
it could only be the same as that given at the beginning of Book v., with 
numbers substituted for "homogeneous magnitudes " and "in respect of size" 
{njXiKonp-a) omitted or altered. We do not find that Nicomachus and the 
rest give any substantially diflerent definition of a ra/io between numbers. 
Theon of Smyrna says, in fact (p, 73, 16), that " ratio in the sense of 
proportion (K6yoi 4 itar' ara'Xoyok) is a sort of relation of two homogeneous 
terms to one another, as for example, double, triple." Similarly Nicomachus 
says {11. II, 3) that "a ratio is a relation of two terms to one another," the word 
for " relation " being in both cases the same as Euclid's {cr)(i<nt}. Theon of 
Smyrna goes on to classify ratios as greater, less, or equal, i.e. as ratios of greater 
inequality, less inequality, or equality, and then to specify certain arithmetical 
ratios which had special names, for which he quotes the authority of Adrastus. 
The names were iroXXaTrXoo'tof, irtfioptoif ivifitp^, iroAAairAofrtcn^topiof} 
iroXXar\iiurt*rtfAM(«j<! (the first of which is, of course, a multiple, while the rest 
are the equivalent of certain types of improper fractions as we should call 
them), and the reciprocals of each of these described by prefixing vini or fui. 
After describing these particular classes of arithmetical ratios, Theon goes on 
to say that numbers still have ratios to one another even if they are different 
from all those previously described. We need not therefore concern ourselves 
with the various types ; it is sufficient to observe that any ratio between 
numbers can be expressed in the manner indicated in Euclid's definition of 
arithmetical proportion, for the greater is, in relation to the less, either one or 
a combination of more than one of the three things, (i) a multiple, {3} a 
submultiple, (3) a proper fraction. 

It is when we come to the definition of proportion that we begin to find 
differences between Euclid, Nicomach us, Theon and lamblichus, " Proportion," 
says Theon (p. 81, 6), " is similarity or sameness of more ratios than one," 
which is of course unobjectionable if it is previously understood what a ratio 
is ; but confusion was brought in by those {like I'hiasyllus) who said that 
there were tkret proporiitms (aVoXo^iai), the arithmetic, geometric, and 
harmonic, where of course the reference is to arithmetic, geometric and 
harmonic means (litcronfut). Hence it was necessary to explain, as Adrastus 
did (Theon, p. 106, 15), that of the several mtatts "the geometric was called 
both proportion /ar extelience and primary... though the other means were 
also commonly called proportions by some writers." Accordingly we have 
Nicomachus trying to extend the term " proportion " to cover the various 
meam as well as a proportion in three or four terms in the ordinary sense. He 
says (it. »i, 3): " Proportion, /ar «ciy/An« (kv/jhus), is the bringing together 
(<ni>VXt)^i;) to the same (point) of two or more ratios \ or, more generally, (the 
bringing tc^ether) of two or more relations {tr^ifn^, even though they be 
subjected not to the same ratio but to a difference or some other (law)." 
lamblichus keeps the senses of the word more distinct. He says, like Theon, 
that " proportion is similarity or sameness of several ratios " (p. 98, 14), and 
that " it is to be premised that it was the geometrical (proportion) which the 
ancients called proportion par excelknct^ though it is now common to apply 
the name genemlly to all the remaining means as well " (p. 100, 15). Pappus 



vit. DEFF. 20— ij] NOTES ON DEFINITIONS 70—21 393 

remarks {in. p. 70, 17), "A mean differs from a proportion in this respect tha^ if 
anything is a proportion, it is also a mean, but not conversely. For there are 
three means, of which one is arithmetic, one geometric and one harmonic." 
The last remark implies plainly enough that there is only one proportion 
(d™Aoy«i) in the proper sense. So, too, says lamblichus in another place 
(p. 104, 19): "the second, the geometric, mean has been called proportion 
par excclletue because the terms contain the same ratio, being separated 
according to the same proportion (aVcl tov airov Xoyov Swaron-t?)." The 
natural conclusion is that of Nesselmann, that originally the geometric 
proportion was called ivakoyia, the others, the arithmetic, the harmonic, etc, 
tneans ; but later usage had obliterated the distinction. 

Of proportions in the ancient and Euclidean sense Theon fp, 82, 10) 
distinguished the continuous (trui-tjf^!) and the separated (Zijjpijfi.iv'ri), using the 
same terms as Aristotle {Eth. I^t'c. 1131 a 32). The meaning is of course 
clear : in the continuous proportion the consequent of one ratio is the ante- 
cedent of the next ; in the separated proportion this is not so. Nicomachus 
(11. 21, 5 — 6) uses the words (onnected (<rwrijLit.ivi}) and disjoined (Snitvyfiivi)) 
respectively. Euclid r^ularly speaks of numbers in continuous proportion as 
" proportional in order, or successively " {ifij« liraXoyoi'). 



Definition 21. 

'QfioiOi ^lirtBoi KoX <rr€pto\ afii$ftOL ihnv al drvdXoyov c^orrc; ras rXcupa^. 

Theon of Smyrna remarks (p, 36, 12) that, among plane numbers, at/ 
squares are similar, while of lTtpofi-^iif!.<! those are similar " whose sides, that 
is, the numbers containing them, are proportional." Here irfpofirfKiji must 
evidently be used, not in the sense of a number of the form n{n + i), but, as 
synonymous with irpofHjieij;, any oblong number ; so that on this occasion 
Theon follows the terminology of Plato and (according to lamblichus) of 
Euclid. Obviously, if the strict sense of rrfpo/ufitiji is adhered to, no two 
numbers of that form can be similar unless they are also c^uat. We may 
compare lamblichus' elaborate contrast of the square and the irfpo/tijuj^. 
Since the two sides of the square are equal, a square number might, as he 
says {p. 8a, 9), be fitly called ISio^iJxj;! (Nicomachus uses rauro^jjiojt) in 
contrast to iripoiiTJinji ; and the ancients, according to him, called square 
numbers " the same " and " similar " (rajirou; re nai ofiotavt), but iTtpo/ii^KtK 
numbers " dissimilar and other " (ovo^oi'ous koI Baripov^), 

With regard to solid numbers, Theon remarks in like manner (p. 37, 2) 
that atl cube numbers are similar, while of the others those are similar whose 
sides are profwrtional,- i.e. in which, as length is to length, so is breadth to 
breadth and height to height. 

DEFmiTION 22. 

Theon of Smyrna (p. 45, 9 sqq.) and Nicomachus (i. 16) both give 
the same definition of a perfect number, as well as the law of formation of 
such numbers which Euclid proves in the later proposition, ix. 36. They 
add however definitions of two other kinds of numbers in contrast with it, 
(i) the oTier-pnfect (wVfpT^Aijv in Nicomachus, virtprtXttot in Theon), the 



m - .. BOOK VII . ;,,...,, 



•r* ;-I Ir-.V 



sum of whose parts, i.e. submultiples, is greater than the numbttr itself, e.g. 1 1, 
24 etc., the sum of the parts of 12 being 6+4 + 3 + 2+1 = 16, and the 
sum of the parts of 34 being 12 + 8 + 6 + 4 + 3 + 2 + i = 36, (2) the defective 
(AAiTTiTv), the sum of whose parts is less than the whole, e.g. S or 14, the 
parts in the first case adding up to 4 + a + i, or 7, and in the second case to 
7 + 2 + I, or 10. All three classes are however made by Theon subdivisions 
of numbers in general, but by Nicouiachus subdivisions of even numbers. 

The term perfect was used by the Pythagoreans, but in another sense, of 
10; while Theon tells us (p. 46, 14) that 3 was also called perfect "because 
it is the first number that has beginning, middle and extremity; it is also both 
a line and a plcme (for it is an equilateral triangle having each side made up 
of two units), and it is the first Imk and potentiality of the solid (for a solid 
must be conceived of in three dimensions)." 

There are certain unexpressed axioms used in Book vii. as there are in 
earlier Books. 

The following may be noted, 

I. If j^ measures B, and ^ measures C, A will measure C. 

a. \l A measures B, apd also measures C, A will measure the difference 
between B and C when they are unequal. 

3, If A measures B, and also measures C, A will measure the sum of B 
and C. 

It is clear, from what we know of the Pythagorean theory of numbers, of 
musical intervals expressed by numbers, of difTerent kinds of means etc., that 
the substance of Euclid Books vii.— ix. was no new thing but goes back, at 
least, to the Pythagoreans. It is well known that the mathematics of Plato's 
Jlmaeus is essentially Pythagorean. It is therefore a priori probable (if not 
perhaps quite certain) that Plato irvSayopi'iet even in the passage (32 a, a) where 
he speaks of numbers " whether solid or square " in continued proportion, 
and proceeds to say that between planes one mean suffices, but to connect 
two solids two means are necessary. This passage has been much discussed, 
but I think that by " planes " and " solids " Plato certainly meant square and 
iolid ntim&ers respectively, so that the allusion must be to the theorems 
estabtished in Eucl. viii. ti, 12, that between two square numbers there is 
one mean proportional number, and between two cube numbers there are 
two mean proportional numbers'. 

..-1. . . Iv 

' It is true that similar p!ine and solid numbers have the $arne property (Eucl, viii. 18, 
19) ; but, if Plato had meant similar pkne and solid numbers generally, I think il would 
have been necessary to specify that they were " similir," whereas, seeing that the Timams a 
as a whole concerned with regular fi|,nires, there is nothing unnatural in allowing rc^lar or 
equilaleral to be understood. Further Plato speaks first of Juni/ini and iytoi and then of 
"planes" [tTrlriia) and "solids" (Fttpti.) in such a way as to surest that ixiviiua cor- 
respoiid to iwlTttSu, and 6n/Kot to vripti. Now the regular meaning m J^a^ii is square (or 
sometimes square rant), and I think it is here used m the sense of sauarr, notwithstanding 
that Plato seems to speak of lAm squares in continued proportion, whereas, in general, the 
mean between two squares as eitremes would not be square but olJong. And, if Suti/uu are 
squares, it is reasonable to suppose that the tyKot afe also ei/iiiiateral, i.e. the "sulids" are 
cubes. 1 am aware that Tb. Habler (Biilisthtia Malkimatita, VIII3, 1008, pp. 173—4) 
thinks that [he passage is to be explained by reference to the problem of the duplication of 
the cube, and does not refer to numbers at all. Against this we have to put the evidence of 
Nicomachus (It. 54, 6) who, in speaking of "a certain Platonic theorem," quotes the very 
same results of Eud. VIii. 11, n. Secondly, it is worth noting that Hiiblet's explanation is 
dulinctly raled out by Democritus tb« Platonjst (jrd cettt. A,D.} who, according to Proclus 



HISTORICAL NOTE i^S 

It is no less clear that, in his method and line of argument, Euclid was 
following earlier models, though no doubt making improvements in the ex- 
position. The tract on the &ciio Cnn&nis, KOTaro/ii; KavoviK (as to the genuine- 
ness of which see above, Vol. !., p. 17) is in style and in the form of the 
propositions generally akin to the Ekmenis. In one proposition (2) the author 
says "ife learned ((ftaSo^ei-) that, if as many numbers as we please be in (con- 
tinued) proportion, and the first measures the last, the first will also measure 
the intermediate numbers " ; here he practically quotes EUm. viii. 7- In the 
3rd proposition he proves that no number can be a mean between two 
numbers in the ratio known as hrmopitu, the ratio, that is, of » ■(■ i to n, where 
« is any integer greater than unity. Now, fortunately, Boethius, De vistitufione 
viuska. III. 1 1 {pp. 885—6, ed. Friedlein), has preserved a proof by Archytas 
of this same proposition ; and the proof is substantially identical with that 
of Euclid. The two proofs are placed side by side in an article by Tannery 
{Sibliothcta Mathematiea, vr,, 1905/6, p. 227). Archytas writes the smaller 
term of the proportion first (instead of the greater, as Euclid does). Let, he 
says, ^, ^ be the " superparticularis proportio " (iu-ifiopio!' hasmfii-a in Euclid). 
Take C, £>£ the smallest numbers which are in the ratio of A to B. [Here 
DE means D + E: and in this respect the notation is different from that of 
Euclid who, as usual, takes a line DF divided into two parts at G, GF 
corresponding to E, and DG to D, in Archytas' notation. The step of taking 
C, DE, the smallest numbers in the ratio of A to B, presupposes Eucl. vii. 
33 J Then DE exceeds C by an aliquot part of itself and of C [cf the 
definition of iTcifiopuK dpi^/ia^ in Nicomachus, i. 19, i]. Let D be the excess 
[i.e. E is supposed equal to C]. " I say that D is not a number but an uniL" 

For, if Z" is a number and a part of DE, it measures JJE ; hence it 
measures E, that is, C- Thus JD measures both C and BE, which is 
impossible ; for the smallest numbers which are in the same ratio as any 
numbers are prime to one another. [This presupposes Eucl, vei. 2a.] There- 
fore -D is an unit ; that is, DE exceeds C by an unit. Hence no number can 
be found which is a mean between two numbers C, DE. Therefore neither 
can any number be a mean between the original numbers A, B which are in 
the same ratio [this implies Eucl. vii. 20]. 

We have then here a clear indication of the existence at least as early as 
the date of Archytas (about 430 — 365 B.C.) of an Eitments of Aritkmetie in 
the form which we call Euclidean ; and no doubt text-books of the sort 
existed even before Archytas, which probably Archytas himself and Eudoxus 
improved and developed in their turn. 

{In P!at<snis Tiaamm (ommtntaria, [+9 c), said that the dlfficultiep of the passage of the 
TitHtuui tiad misUd some people into connecting it with tbe duplication of the cube, 
whereas it really referred to similar planes and solids with sides in ra/iotial numbers^ 
Thirdly, I do not think that, under tht supposition that the Delian problem is referred to, 
we get the required sense. The problem in that case is not that of finding two mean 
proportionals Between two eudii Ijut that of finding a second cybe the content of which 
ahall Ue equal to twice, or k times (where ^ is any numtier not & complete cube), the content 
of a given Oi\x (^. Two mean proportionals are found, not between cubes, but between 
two siraight linn in the ratio of 1 to k, or between a and ks. Unless .( is a culje, there 
would lie no point in saying that two means are necessary to connect t aad k, and not one 
mean ; for ijk is no more naluial than .Ji, and would be less natural in the case where * 
happened to t>e square. On the other hand, if ^ is a cube, ^ that it Is a question of finding 
means between tuhe numbers, the dictum of Plato is perfectly intelligible ; nor is any real 
difficulty caused by the generality of the statement that two means are al-ivays necessary to 
connect them, because any property enunciated generally of two cut>e numbers should 
obviously be true of cubes 9S sikH, that is, it must hold in the extreme ease of two cubes 
which are ^me to aw att&ihir. 



BOOK VII. PROPOSITIONS. 



A 
H 



F ■ 



O 
H 



Proposition i. 

Two UHsqtiai numbers being set out, and the less being 
continually subtracted in turn from the greater, if the number 
which is left never measures the one before it until an unit is 
left, the original numbers mill be prime to one another. 

For, the less of two unequal numbers AB, CD being 
continually subtracted from the greater, let the 
number which is left never measure the one 
before it until an unit is left ; 

I say that AB, CD are prime to one another, 
that is, that an unit alone measures AB, CD. 

For, liAB, CD are not prime to one another, 
some number will measure them. 

Let a number measure them, and let it be 
E\ let CD, measuring BF, leave FA less than 
itself, 

let AF, measuring DG, leave GC less than Itself, 

and let GC, measuring FH, leave an unit HA. 

Since, then, E measures CD, and CD measures BF, 
therefore E also measures BF. 

But it also measures the whole BA ; 
therefore it will also measure the remainder ..<^^. 

But AF measures DG ; 
therefore E also measures DG. 



VII. i] PROPOSITION I *j$)l 

But it also measures the whole DC ■ 
therefore it will also measure the remainder CG. 

But CG measures FH ; 
therefore E also measures FH. 

But it also measures the whole FA ; 

therefore it will also measure the remainder, the unit AH, 
though it is a number : which is impossible. 

Therefore no number will measure the numbers ^5, CD; 
therefore AB, CD are prime to one another. [vn. Def. i a] 

Q. E, D. 

It is proper to remark here that the representation in Books vn. to ix. of 
numbers by straight lines is adopted by Heiberg from the mss. The method 
of those editors who substitute poirtti for lines is open to objection because it 
practically necessitates, in many cases, the use of specific numbers, which is 
contrary to Euclid's manner. 

"Let CD, measuring BF, leave FA less than itself." This is a neat 
abbreviation for saying, measure along BA successive lensths equal to CD 
until a point F is reached such that the length FA remaining is less than 
CD ; in other words, let BF be the largest eitact multiple of CD corstained 
in BA. 

Euclid's method in this proposition is an application to the particular 
case of prime numbers of the method of finding the greatest common measure 
of two numbers not prime (t) one another, which we shall find in the next 
proposition. With our notation, the method may be shown thus. Supposing 
the two numbers to be a, b, we have, say, 

t ' 

If now a, 3 are not prime to one another, they must have a commcm 
measure t, where e is some integer, not unity. 

And since t measures a, 6, it measures a ~pb, i.e. (, 

Ag^n, since t measures b, c, it measures 6-fc, i.e. d, 

and lastly, since * measures <■, d, it measures e~rd, i.e. i; ' i . 1. if 

which is impossible. 

Therefore there is no integer, except unity, that measures a, b, which are 
accordingly prime to one another. 

Observe that Euclid assumes as an axiom that, if a, b are both divisible by 
f, so is a -pb. In the next proposition he assumes as an axiom that c will in 
the case supposed divide a +pb, 



c 

■F 



»98 BOOK Vn [vu. 1 



Proposition 2. 

Given (wo numbers not prime to one another, to find their 
greatest contnion measure. 

Let AB, CD b« the two given numbers not prime to one 
another. 

Thus it is required to find the greatest J^ 
common measure of AB, CD. 

If now CD measures AH — and it also ^ 
measures itself — CD is a common measure of 
CD, AB. 

And it is manifest that it is also the greatest ; 
for ao greater number than CD will measure ^ ^ 
CD. 

But, if CD does not measure AB, then, the less of the 
numbers AB, CD being continually subtracted from the 
greater, some number will be left which will measure the one 
before it. 

For an unit will not be left ; otherwise AB, CD will be 
prime to one another [vii. i], which is contrary to the 
hypothesis. 

Therefore some number will be left which will measure 
the one before it. 

Now let CD, measuring BE, leave EA less than itself, 
let EA, measuring DF, leave EC less than itself, 

and let CA" measure AE. 

Since then, CF measures AE, and AE measures DF^ 
therefore CF will also measure DF. 

But it also measures itself; 
therefore it will also measure the whole CD. 

But CD measures BE ; 
therefore C/^also measures BE. 

But it also measures EA ; 
therefore it will also measure the whole BA. 

But it also measures CD ; 
therefore CF measures AB, CD. 

Therefore CF'is a common measure oi AB, CD, 



VII. 2] PROPOSITION 2 -4$9 

I say next that it is also the greatest. 

For, if CF is not the greatest common measure of AB, 
CD, some number which is greater than CF will measure the 
numbers AB, CD. 

Let such a number measure them, and let it be G. 

Now, since G measures CD, while CD measures BE, 
G also measures BE. "'" 

But it als© measures the whole BA ; 

therefore it will also measure the remainder AB, 

But .^^ measures /?y^; 
therefore G will also measure DF. 

But it also measures the whole DC ; 

therefore it will also measure the remainder CF, that is, the 
greater will measure the less : which is impossible. 

Therefore no number which is greater than Ci^ will measure 
the numbers AB, CD ; 

therefore CF is the greatest common measure of AB, CD, 

PORISM. From this it is manifest that, if a number 
measure two numbers, it will also measure their greatest 
common measure. Q. E. D.. 

Here we have the exact niethod of finding the greatest common measure 
given in the text- books of algebra, including the reductio ad abmrdum proof 
that the number arrived at is not only a common measure but the greatest 
common measure. The process of finding the greatest common measure 
is simply shown thus : 

P± 
€)biq •■ -. 

We shall arrive, says Euclid, at some number, say d, which measures the one 
before it, i.e. such that e = rd. Otherwise the process would go on until we 
arrived at unity. This is impossible because in that case a, b would be prime 
to one another, which is contrary to the hypothesis. 

Next, like the text-books of algebra, he goes on to show that d will be some 
common measure of a, b. For d measures c ; 
therefore it measures jff + i^, that is, #, ■ ' 

and hence it measures pb + ^, that is, a. 

Lastly, he proves that d is the greatest common measure of a, b as follows. 

Suppose that ? is a common measure greater than d. 

Then e, measuring a, i, must measure a-fb, or c. • 



Soo BOOK VII [vii. *, 3 

Similarly « must measure 6 -qe, that is, d: which is impossible, since e is 

by hypothesis greater than d. . . i - • . ; 

Therefore etc. " .-! - . 

Euclid's proposition is thus identical with the algebrmical proposidon as 
generally given, e.g. in Todhunter's algebra, except that of course Euclid's 
numbers are integers. 

Niconiachus gives the same rule (though without proving it) when he 
shows how to determine whether two given odd numbers are prime or not 
prime to one another, and, if they are not prime to one anothet, what is their 
common measure. We are, he says, to compare the numbers in turn by 
continually taking the less from the greater as many times as possible, 
then taking the remainder as many times as jKwsible from the less of the 
original numbers, and so on ; this process " will finish either at an unit or at 
some one and the same number," by which it is implied that the division of a 
greater number by a less is done by separate snbfraetiens of the less. Thus, 
with regard to 2 1 and 49, Nicomachus says, " I subtract the less from the 
greater ; a8 is left ; then ^;ain I subtract from this the same 2 r (for this is 
possible); 7 is left; I subtract this from 2t, 14 is left; from which I again 
subtract 7 (for this is possible); 7 will be left, but 7 cannot be subtracted from 
7." The last phrase is curious, but the meaning of it is obvious enough, as 
also the meaning of the phrase about ending " at one and the same number." 

The proof of the Porism is of course contained in that part of the propo- 
sition which proves that G, a common measure different from CF^ must 
measure CF. The supposition, thereby proved to be false, that G is greater 
than CFdxxA not affect the validity of the proof that G measures CF'xn any 
case. 



Proposition 3. 

" ■. Given three numbers noi prime to one another, to find their 
greatest common measure. 

Let A, B, C be the three given numbers not prime to 
one another ; 

thus it is required to find the greatest 
common measure oi A, B, C. 

For let the greatest common measure, 
D, of the two numbers .^, .5 be taken ; 

[vu. l] 

then D either measures, or does not 
measure, C, 

First, let it measure iL 

But it measures A, B also ; 
therefore D measures A, B, C ; 
therefore /? is a common measure of .^, B, C, 

I say that it is also the greatest. 



E| Fl 



VII. 3] PROPOSITIONS 2, 3 30I 

For, if i? is not the greatest commoQ measure of^, B, C, 
some number which is greater than D will measure the numbers 

A, B, C. 

Let such a number measure them, and let it be E. 
Since then E measures A, B, C, _ , 

it will also measure A, B ; 

therefore it will also measure the greatest common measure 
of A, B. [vii. 2, For.] 

But the greatest common measure of A, B is D ; 
therefore E measures D, the greater the less : which is 
impossible. 

Therefore no number which is greater than Z? will measure 
the numbers A, B, C; 

therefore D is the greatest common measure of A, B, C. 

Next, let D not measure C ; 

I say first that C, D are not prime to one another. 

For, since A, B, C are not prime to one another, some 
number will measure them. 

Now that which measures A, B, C will also measure A, 

B, and will measure D, the greatest common measure oi A, B, 

[vn. %, Por.] 
But it measures C also ; ^ 

therefore some number will measure the numbers D, C\ 

therefore D, C are not prime to one another. 

Let then their greatest common measure E be taken. 

[vii. i\ 
Then, since E measures D, 

and /? measures ^, ^, • 

therefore E also measures A, B. , • 

But it measures C also ; ' • 

therefore E measures A, B, C; , 

therefore .£" is a common measure of A, B, C, 

I say next that it is also the greatest. 

For, if E is not the greatest common measure of A, B, C, 
some number which is greater than E will measure the 
numbers A, B, C. 

Let such a number measure them, and let it be F. 



$oz BOOK VII • [vii. 3 

' Now, since /^measures A, B, C, ■ u * '-i. '• ' 
it also measures /^, Z? ; ' ' -' ' >.i". 

therefore it will also measure the greatest common measure 
of W, B. [vn. 2, Por.] 

But the greatest common measure of A, B is D ; 
therefore J^ measures D. 

And it measures C also ; ' ' '"' . '' 

therefore /^measures Z>, C; 

therefore it will also measure the greatest common measure 
of D, C. [vn. i, Por.] 

But the greatest common measure of Z?, C is ^ ; 
therefore /" measures £, the greater the Jess: which is 
impossible. 

Therefore no number which is greater than S will measure 
the numbers A, B, C; 
therefore £ is the greatest common measure of A, B, C, 

Q. E. D. 

Euclid's proof is here longer than we should make it because he 
distinguishes two cases, the simpler of which is really included in the other- 
Having taken the greatest common measure, say d, of a, t>, two of the 
three given numbers a, b, c,h^ distinguishes the cases 

(i) in which d measures (, 

(2) in which d does not measure c. 

In the first case the greatest common measure of d, e'\^ d itself; in the 
second case it has to be found by a repetition of the process of v 11. 2. In 
either case the greatest common measure oi a, 6, ( is the greatest common 
measure of d, c. 

But, after disposing of the simpler case, Euclid thinks it necessary to 
prove that, if d does not measure c, d and ( must necessarily have a greatest 
common measure. This he does by means of the original hypothesis that 
tf, b, c are not prime to one another. Since they are not prime to one another, 
they must have a common measure; any common measure of a, * is a measure 
of d, and therefore any common measure of a, *, r is a common measure of 
d, c \ hence d, c must have a common measure, and are therefore not prime to 
one another. 

The proofs of cases (i) and (2) repeat exactly the same alignment as we 
saw in vii. 3, and it is proved separately for <f in case (i) and t in case (3), 
where < is the greatest common measure of d, (, 

(a) that it is a common measure of a, b, e, 

(fi) that it is the greaiesi common measure. 

Heron remarks (an-Nairlzi, ed. Curtze, p. 191) that the method does 
not only enable us to find the greatest common measure of /Arte numbers ; 
it can be used to find the greatest common measure of as many numbers 



Miie 



VII. 3, 4l PROPOSITIONS 3, 4 303 

as we please. This is because any number measuring two numbers also 
measures their greatest common measure ; and hence we can find the g.c.m. 
of pairs, then the g.c.m, of pairs of these, and so 00, until only two numbers 
are lefl and we find the g.c.m. of these. Euchd tacitly assumes this extension 
in VII. 33, where he takes the greatest common measure ofay many numbers 
as we phase. 

Proposition 4. 

Any number is either a pari or parts of any number, the 
less of the greater. 

Let A, BC be two numbers, and let BC be the less ; \ 
I say that BC is either a part, or parts, of A. 

For A, BC are either prime to one another 
or not. 

First, let A, BC be prime to one another. 

Then, if BC be divided into the units in it, 
each unit of those in BC will be some part of A ; 
so that BC is parts of ^. 

Next let A, BC not be prime to one another; 
then .5C either measures, or does not measure, A. 

If now BC measures A, BC is a part oi A, 

But, if not, let the greatest common measure D of A, BC 
be taken ; [vn. 2] 

and let BC be divided into the numbers equal to D, namely 
BE, EF, FC. 

Now, since D measures ^4, /? is a part of A. 

But D is equal to each of the numbers BE, EF, FC; -i* 
therefore each of the numbers BE, EF, FC is also a part of A ; 
so that BC is parts of A. 

Therefore etc. 1 . 1 

The meaning of the enunciation is of course that, if a, b be two numbers 
of which i is the less, then b is either a submultiple or soirn proper fraction of a. 

(i) If a, b are prime to one another, divide each into its units ; then b 
contains b of the same parts of which a contains a. Therefore b is " parts " or 
i. proper Jradion of a. 

(i) If a, b be not prime to one another, either b measures a, in which 
case i is a submuttiple or " part " of a, or, if ^ be the greatest common 
measure of a, b, we may put a — mg and h=ng, and h will contain « of the 
same parts (£) of which a contains m, so that b is again "parts," or a. proper 
fraction, of a. 



$04 BOOK VII [vii. 5 

■ I , Proposition 5. 

If a number be a part of a number, and another be the 
same part of another, the sum will also be the same part of the 
sum that the one is of the one. 

For let the number -(4 be a part of BC, 

and another, D, the same part of another EF that A is of ^C; 

I say that the sum of A, D '\% also the same 
part of the sum of BC, EF that A is of BC. 

For since, whatever part A is of BC, D 
IS also the same part oi EF, 

therefore, as many numbers as there are in 
j5C equal to A, so many numbers are there 
also in EF equal to D. 

Let BC be divided into the numbers equal to A, namely 
BG, GC, 

and EF into the numbers equal to D, namely EH, HF\ 

then the multitude of BG, GC will be equal to the multitude 
of EH, HF. 

And, since BG is equal to A, and EH to D, 
therefore BG, EH are also equal to A, D. 

For the same reason 
GC, HF 2iX^ also equal to A, D. 

Therefore, as many numbers as there are in BC equal to 
A, so many are there also in BC, £"7^ equal to A, D. 

Therefore, whatever multiple BC \^o\ A, the same multiple 
also is the sum of BC, EF of the sum of A, D. 

Therefore, whatever part A is of BC, the same part also 
is the sum of A, D of the sum of BC, EF. 



Q. E. D. 



If a = -6, and e = -d, then 



it 
The proposition is of course true for any quantity of pairs of numbers 
similarly related, as is the next proposition slso ; and both propositions aie 
used in the extended form in vil 9, 10. 



A 








C 


D 


Q 




H 


B 




E 



VII. 6J PROPOSITIONS 5, 6 305 

Proposition 6. 

// a number be parts of a number, and another be the same 
parts of another, (he sum will also be the same parts of the sum 
that the one is of the one. 

For let the number AB be parts of the number C, 
and another, DE, the same parts of another, 
F, that AB is of C ; 

I say that the sum of AB, DE is also the 
same parts of the sum of C, F that AB is 
of C 

For since, whatever parts AB is of C, 
DE is also the same parts of F, 
therefore, as many parts of C as there are 
in AB, so many parts of /^ are there also in DE. 

Let AB be divided into the parts of C, namely AG, GB, 
and DE into the parts of ^, namely DH, HE; 
thus the multitude of AG, GB will be equal to the multitude 
of Z?jy, HE, 

And since, whatever part AG is of C, the same part is 
/JiYof Aalso, 

therefore, whatever part AG\%o{ C, the same part also is the 
sum of AG, DH of the sum of C, F. [vii. 5] 

For the same reason, ^ ' ■ . ; 

whatever part GB is of C, the same part also js the sum of 
GB, HE of the sum of C, F, 

Therefore, whatever parts AB is of C, the same parts also 
is the sum of AB, DE of the sum of C, F. 



Q. E. D. 



If a = — i, ana c = — d, 

r n n 

then a + €=- (fi + d). 



More generally, if 



OT . Iff V fft f 

a = ~ b, c= — a, «-- /, 
n ' ft n 



then {a +(+ ( + g+ ,,,) = - {i-*-t/+/-i 



jo6 BOOK VII [vir. 6, ^ 

In Euclid's proposition m<.n, but tht generality of the result is of course 
not aiTected. This proposition and the last are complementary to v, i, which 
proves the corresponding result with multiple substituted for "pari" or 

Proposition 7, r 

If a number be that part of a number, which a number 
subtracted is of a number subtracted, the remainder will also 
be the same part of the remainder thai the whole is of the 
whole. 

For let the number AB be that part of the number CD 
which AE subtracted is of CF subtracted ; 

I say that the remainder EB is also the same part of the 
remainder FD that the whole AB is of the whole CD. 



For, whatever part AE is of CF, the same part also let 
EB be of CG. 

Now siQce, whatever part AE is of CF, the same part 
also is EB of CG, 

therefore, whatever part AE is of CF, the same part also is 
AB of GF. [vii. s] 

But, whatever part AE is of CF, the same part also, by 
hypothesis, is AB of CD ; 

therefore, whatever part AB \% o^ GF, the same part is it of 
CD also ; ,. . 

therefore GF is equal to CD. 

Let CF be subtracted from each ; 
therefore the remainder GC is equal to the remainder FD. 

Now since, whatever part AE is of CF, the same part 
also is EB of GC, 

while GC is equal to FD, 

therefore, whatever part AE is of CF, the same part also is 
^^of/'Z>. 

But, whatever part AE is of CF, the same part also is AB 
oiCD; 



vii. 7, 8] PROPOSITIONS 6~8 307 

therefore also the remainder EB is the same part of the 
remainder FD that the whole AB is of the whole CD. 



Q. E. D, 

If a=^ -li and f=-rf, we are to prove that 

a~c = -(6-d), 

a result differing from that of vji. 5 in that minus is substituted for /Ar. 
Euclid's method is as follows. 

Suppose that e is taken such that 

a-^ = -e. (i) 

Now e=-d. 

n 

Therefore a = -{d-¥e), [vii. 5] 

whence, from the hypothesis, d-ve = b, 
so that e = b-d, 

and, substituting this value of^ in (i), we have 

.i , a-e=^-{b'-d). 



Proposition 8. 

If a number be the same parts of a number that a number 
subtracted is of a number subtracted, the remainder will also 
be the same parts of the remainder that the whole is of the 
whole. 

For let the number AB be the same parts of the number 
CD that AE subtracted is of CF 
subtracted ; 

I say that the remainder EB is 
also the same parts of the re- 
mainder FD that the whole AB 
is of the whole CD. 

For let GH be made equal to AB, 

Therefore, whatever parts GH is of CD^ the same parts 
also is AE of CF. 

Let GHhe, divided into the parts of CD, namely GK, KH, 
and AE into the parts of CF, namely AL, LE; 

thus the multitude of GK, KH-^\\\ be equal to the multitude 
of AL, LE. 



c 




f = 


Q 


M K 


N H 




A 


L 


t B 



3A8 book vn ' [vii. 8 

Now since, whatever part GK is of CD, the same part 

also is ^Z of C/s '■"-<• 

while CD is greater than CF, 
therefore GK is also greater than AL, 

Let GM be made equal to AL. 

Therefore, whatever part GK is of CD, the same part also 
is GM oi CF; 

therefore also the remainder MK is the lame part of the 
remainder FD that the whole GK is of the whole CD. [vii. 7] 

Again, since, whatever part KH is of CD, the same part 
also is EL of CF, 
while CD is greater than CF", 
therefore HK is also greater than EL. 

Let KN be made equal to EL . 

Therefore, whatever part KN is of CD, the same part 
also is KJV of CF; 

therefore also the remainder JV/f is the same part of the 
remainder FD that the whole A'// is of the whole CD, 

[vn. 7] 

But the remainder MK was also proved to be the same 
part of the remainder FD that the whole GK is of the whole 
CD; 

therefore also the sum of MK, NH is the same parts of DF 
that the whole HG is of the whole CD. 

But the sum of MK, NH is equal to EB, 
and HG is equal to BA ; 

therefore the remainder EB is the same parts of the remainder 
FD that the whole AB is of the whole CD. 





Q. E. D. 


It - - «--<( and i==~d. 


{m <n) 


then fl-f=^(i-i). 




Euclid's proof amounts to the following. 




Take e equal to - b, and /equal to - d. 

ft H 




Then since, by hypothesis, b->d. 




and, by VII. 7, e -f- -{b- d). 

fi 





vii. H, 9J PROPOSITIONS 8, 9 309 

Repeat this for all the parts equal to t and/that there are in a, b respec- 
tively, and we have, by addition {a, b containing m of such parts respectively), 

»'i.:~n = "'-{b-i). 

But m{i—/) = a-c. 

Therefore a-e- (d-d). 

n 

The propositions v[i, 7, 8 are complementary to v. 5 which gives the 
corresponding result with multipit in the place of " part " or " parts." 



Proposition 9. 

If a number be a pari of a number, and another be the 
same part of another, alternately also, whatever part or parts 
ike first is of the third, the same part, or the same parts, will 
the second also be of the fourth. 

For let the number A he a part of the number BC, 
and another, D, the same part of another, £/^, .. , 

that A is of BC ; 

I say that, alternately also, whatever part or ^ 

parts A is of D, the same part or parts is BC , a 
oi£F3.\so. ^ ^ 

For since, whatever part A is of BC, the 
same part also is D of £/^, 

therefore, as many numbers as there are in BC equal to A, 
so many also are there in £1^ equal to D. 

Let BC be divided into the numbers equal to A, namely 
BG, GC, 

and ^/^into those equal to D, namely £//, HF\ -w ,„ 
thus the multitude of BG^ GC will be equal to the multitude 
of EH, HF. 

Now, since the numbers BG, GC are equal to one another, 

and the numbers EH, HF are also equal to one another, 

while the multitude of BG, GC is equal to the multitude of 
EH, HF, 

therefore, whatever part or parts BG is of EH, the same 
part or the same parts is GC oi HF also ; 

so that, in addition, whatever part or parts BG is of EH, 
the same part also, or the same parts, is the sum BC of the 
sum £F. [vit. s, 6] 



IflD BOOK VII [vii, 9, to 

But BG is equal to A, and EH to D\ ' • 

therefore, whatever part or parts A is of D, the same part or 
the same parts is BC of EF also. 

Q. E. D. 

If a = - b and e = ~ d, then, whatever fraction {" part " or " parts") a is of 
ft ft 

c, the same fraction will ihe of ti. 

Dividing i into each of its parts equal to a, and d into each of its parts 
equal to i, it is clear that, whatever fraction one of the parts a is of one of the 
parts c, the same fraction is any other of the parts a of any other of the parts ^. 

And the number of the parts a is equal to the number of the parts /:, viz. n. 

Therefore, by vii. 5, 6, na is the same fraction of »c that a is of (, i.e. 6 is 
the same fraction of d that o is of <:. 



Proposition 10. 

// a numier be parts of a number, and another be ike 
same Paris 0/ another, alternately also, wliatever parts or part 
the first is of the third, the same parts or the same part will 
the second also be of t/ie fourth, 

For let the number AB he parts of the number C, 
and another, DE, the same parts of another, 
^; 

I say that, alternately also, whatever parts or 
part AB is of -DE, the same parts or the 
same part is C of E also. 

For since, whatever parts AB is o( C, '^' 
the same parts also is DE of E, 
therefore, as many parts of C as there are 
in AB, so many parts also of E are there in DE. 

Let AB be divided into the parts of C, tiamely AG, GB, 
and DE into the parts of E, namely D//, HE ; 
thus the multitude of AG, GB will be equal to the multitude 
oiDH, HE. 

Now since, whatever part AG xsoi C, the same part also 
IsDHo^E, 

alternately also, whatever part or parts AGx^tS DH, 

the same part or the same parts is C of E also. [vu. 9] 

For the same reason also, 
whatever part or parts GB is of HE, the same part or the 
same parts is C of J^ also ; 



v[[. lo, ii] PROPOSITIONS 9—11 JH 

SO that, in addition, whatever parts or part AB is of /?£', 
the same parts also, or the same part, is C of F> [vii. 5, 6] 

Q. E. D. 

\{ a = — b and e ~ —d, then, whatever fraction a is of c, the same fraction 

is j of d. 

To prove this, a is divided into its m parts equal to t>jn, and c into its 
»J parts equal to djn. 

Then, by v[i. 9, whatever fraction one of the m parts of a is of one of the 
m parts of ^, the same fraction is a of d. 

And, by vii, 5, 6, whatever fraction one of the /// parts of a is of one of 
the t» parts of c, the same fraction is the sum of the parts of a (that is, o) of 
the sum of the parts ol c (that is, i). 

Whence the resliit follows. 

Fn the Greek text, after the words " so that, in addition " in the last line 
but one, is an additional explanation making the reference to vii. 5, 6 cl^rer, 
as follows: "whatever part or parts AG is of DII, the same part or the 
same parts is GS of HE also ; 

therefore also, whatever part or parts ^C is of DH, the same part or the same 
parts is AB of DE also. [vii. 5, 6] 

But it was proved that, whatever part or parts AG \s of DH, the same 
part or the same parts is C of Faho ; 
therefore also " etc, as in the last two lines of the text. 

Heiberg concludes, on the authority of P, which only has the words in 
the margin in a later hand, that they may be attributed to Theon. 

. ■.. Proposition ii. 

If, as whole is to whole, so is a number subtracted to a 
number subtracted, the retnainder will also be to the remainder 
as whole to whole. 

As the whole ^5 is to the whole C£>, so tet A£ subtracted 
be to C/^ subtracted ; 

I say that the remainder £B is also to the remainder 
FD as the whole AB to the whole CD. 

Since, as AB is to C£), so is A£ to CF, 
whatever part or parts AB is of CD, the same part 
or the same parts is AB of CF a.lso ; [vn. Def. 20] 

Therefore also the remainder BB is the same 
part or parts of FD that AB is of CD. [vn, 7, 8] 

Therefore, as £B is to FD, so is AB to CD. [vn. Def. 30] 

Q. E. D. 

It will be observed that, in dealing with the proportions in Props, 11—13, 
Euclid only contemplates the case where the first number is "a part" or 
"parts" of the second, vhile in Prop. 13 he assumes the first to be "a part" 



F 



jtjl BOOK VII [vii. II, 12 

or "parts" of the third also; that is, the first number ts in all three propositions 
assumed to be less than the second, and in Prop. 13 less than the third also. 
Yet the figures in Props. 1 1 and 1 3 are inconsistent with these assumptions. 
If the facts are taken to correspond to the figures in these propositions, it is 
necessary to take account of the other possibilities involved in the definition 
of proportion (vii. Def 20), that the first number may also he a multiple, or 
a multiple //kj "a part" or " parts" (including owe as a multiple in this case), 
of each number with which it is compared. Thus a number of different cases 
would have to be considered. The remedy is to make the ratio which is in 
the lower terms the first ratio, and to invert the ratios, if necessary, in order 
to make " a part "or " parts " literally apply. 

If a : i ^ ^ : d, {a > c, b > tf) 

then {a — c):{b~d) = a;b. 

This proposition for numbers corresponds to v. 19 for magnitudes. The 
enunciation is the same except that the masculine (agreeing with apiS/io!) 
takes the place of the neuter (agreeing with niytdm). 

The proof is no more than a combination of the arithmetical definition of 
proportion (vu. Def. 20) with the results of vii, 7, 8. The language of propor- 
tions is turned into the language of fractions by Def. zo ; the results of vii. 7, 8 
are then used and the language retransformed by Def. 20 into the language of 
proportions. 

Proposition 12. 

If there be as Tnany numbers as we please in proportion, 
then, as one of ike antecedents is to one of the consequents, so 
are all the antecedents to ail the consequents. 

Let A, By C, D be as many numbers as we please in 
proportion, so that, 

as A is to By so is C to Z? ; 
I say that, as .^ is to B, so are A, C to B, D. 

For since, as A is to B, so Is C to D, aI bI c 

whatever part or parts A is of B, the same part 
or parts is C q{ D also. [vn. Def. 20] 

Therefore also the sum of A, C is the same 
part or the same parts of the sum of B, D that A is of B. 

[vii. 5, 6] 

Therefore, as A is to B, so are A, C to B, D. [vn. Def. 20] 

If a:({ = l,:V = c:i;=..., 

then each ratio is equal to {a + ^ + ^+ ...) : (0' + *'+/ + ,..). 

The proposition corresponds to v, 1 2, and the enunciation is word for word 
the same with that of v. 12 except that apiSfio? takes the place of f^iyt&oi. 

Again the proof merely connects the arithmetical definition of proportion 
(vn. Def 20) with the results of vii, 5, 6, which are quotttd as true for any 
number of numbers, and not merely for two numbers as in the enunciations of 
VII. s, 6. 



'3. U] 



PROPOSITIONS 11—14 



313 



Proposition 13. 

If four numbers be proportional, they will also be propor- 
tional alternately. 

Let the four numbers A, B, C, D ha proportional, so that, 
as A is to B, so is C to Z? ; 
I say that they will also be proportional alternately, so that, 
as A is to C, so will B be to D. 
For since, as A is to B, so is C to D, 
therefore, whatever part or parts A is of B, 
the same part or the same parts is C of Z? also. 

[v[i. Def. ao] 
Therefore, alternately, whatever part or 
parts A is of C, the same part or the satne 
parts \5 B oi D also. 

Therefore, as ^ is to C, so is B to D. 



If 



[vii. 10] 
[y\\. Def. so] 
Q. E. D. 



a : b = e : d, 
then, alternately, a : c = b id. 

The proposition corresponds to v. 16 for magnitudes, and the proof 
consists in connecting vii. Def. 20 with the result of vii. 10. 

Proposition 14. 

If there be as many numbers as me please, and others equal 
to them in multitude, which taken two and two are in the same 
ratio, they will also be in the same ratio ex aequali. 

Let there be as many numbers as we please A, B, C, 
and others equal to them in multitude D, E, F, which taken 
two and two are in the same ratio, so ihat, 

as .(4 is to B, so is Z? to ^, 
and, as ^ is to C, so is E to F\ m ■ f . ■ 

I say that, ex aequali, 

as A is to C, so also is D to F. 



"B" 



D 



c — f 

For, since, as A is to B, so is D to E, 
therefore, alternately, 

as A is to D, so is B to E. 



[vn. 13] 



314 BOOK Vir [vii. 14, IS 

Again, since, as B is to C, so is E to F, 
therefore, alternately, 

as B is to Ey so is C to F. [vii. 13] 

But, as J? is to E, so is W to D; 
therefore also, as y^ is to D, so is C to F. 
Therefore, alternately, 

as A is to C, so is D to F. . [iij 

If a: b = d.e, 

and d : c = t -.ft 

tKen, ex aepiali, a ; c = d ./; 

and the same is true however many successive numbers are so related. 

The proof is simphcity itself. 

By VII. 13, alternately, a : il - i : f, 

and b : e = c '. f. . ■ _ 

Therefore a\d = c:f, 1 

and, again alternately, a : c^d if. 

Observe that this simple method cannot be used to prove the corresponding 
proposition for magnitudes, v. 22, although v. 22 has been preceded by the 
tivo propositions in that Book corresponding to the propositions used here, 
viz, V. 16 and v, 1 1. The reason of this is that this method would only prove 
V. 22 for six magnitudes all t>f tht sttmt kind, whereas the magnitudes in v. jj 
are not subject to this limitation. 

Heiberg remarks in a note on V[r. 19 that, while Luclid has proved 
several propositions of Book v. over again, by a separate proof, for numbers, 
he has neglected to do so in certain cases; e.g., he often uses v. 1 1 in these pro- 
positions of Book VII., V. 9 in vii. 19, v. 7 in the same proposition, and so on. 
Thus Heiberg would apparently suppose Euclid "^o use v, 1 1 in the last step 
of the present proof (Raiies whkk art the same with ike same ratio are also the 
same with one another). I think it preferable to suppose that Euclid regarded 
the last step as axiomatic ; since, by the definition of proportion, the first 
number is the same multiple or the same part or the same parts of the second 
that the third is of the fourth : the assumption is no more than an assumption 
that the numbers or proper fractions which arc respectively equal to the same 
number or proper fraction are equal to one another. 

Though the proposition is only proved of six numbers, the extension to as 
many as we please (as expressed in the enunciation) is obvious. 



Proposition i 5. 

If an unit measure any number, and another number measure 
any other number tfie same number of times, alternately also, 
the unit ivill measure the third number the same number of 
times that the second measures the fourth. 



vit. 15] PROPOSITIONS 14. >5 jij 

For let the unit A measure any number BC, 
and let another number D 

measure any other number EF ~^— + !l! — ? 

the same number of times ; 

I say that, alternately also, the ^ f L F 

unit A measures the number 

D the same number of times that BC measures EF. 

For, since the unit A measures the number BC the same 
number of times that D measures /:F, 

therefore, as many units as there are in BC, so many numbers 
equal to D are there in ^^also. 

Let BC be divided into the units in it, BG, GH, HC, 
and EF into the numbers EK, KL, Z./^ equal to D. 

Thus the multitude of BG, GH, HC will be equal to the 
multitude of EK, KL, LF. 

And, since the units ^C GH, HCzx^ equal to one another, 

and the numbers EK, KL, LF are also equal to one another, 

while the multitude of the units BG, GH, HC is equal to the 
multitude of the numbers EK, KL, LF, 

therefore, as the unit BG is to the number EK, so will the 
unit GH be to the number KL, and the unit HC to the 
number LF, 

Therefore also, as one of the antecedents is to one of 
the consequents, so will all the antecedents be to all the 
consequents ; [vii. iz] 

therefore, as the unit BG is to the number EK, so is BC to 
EF. 

But the unit BG is equal to the unit A, 

and the number EK to the number D. 

Therefore, as the unit A is to the number D, so is BC to 
EF. 

Therefore the unit A measures the number D the same 
number of times that BC measures EF. Q. E. d. 

If there be four numbers \,m,a, ma (such that i measures m the same 
number of times that a measures ma), i measures a the same number of 
times that m measures ma. 

Except that the first number is unity and the numbers are said to tfieasure 
instead of being a ^art of others, this proposition and its proof do not differ 
from VII, 9 J in fact this proposition is a particular case of the other. 



31« ■. BOOK VII 1 i" [vii. 1 6 

Proposition i6. 

// two numbers by multiplying one another make certain 
numbers, the numbers so produced mill be equal to one another. 

Let A, B he two numbers, and let A by multiplying B 
make C and ^ by multiplying ,j. 

A make D ; f^ 

I say that C is equal to D, b 

For, since A by multiply- c — ■ 

ing B has made C, o 

therefore B measures C ac- — % 
cording to the units in A. 

But the unit E also measures the number A according to 
the units in it ; 

therefore the unit E measures A the same number of times 
that B measures C. 

Therefore, alternately, the imit E measures the number B 
the same number of times that A measures C. [vn. ij] 

Again, since B by multiplying A has made D, 

therefore A measures D according to the units in ^. ' ' 

But the unit E also measures B according to the units 
in it ; 

therefore the unit E measures the number B the same 
number of times that A measures D. 

But the unit E measured the number B the same number 
of times that A measures C ; 

therefore A measures each of the numbers C, D the same 
number of times. 

Therefore C is equal to i?. ' q. e. d. 

■1. The numbers >□ produced. The Gieek hu al yoituw i( niriir, " the (numbers) 
produced /rffjw fJutJt." By *'from them'* Euclid means "from the original numbers," though 
this is not very clear even in the Greek. I think ambiguity is best avoided by leaving out 
the words. 

This proposition proves that, if any nttmbers bt fmtttiplied together, the order 
of muUiplication is indifferent, ox ab-ba. 

It '\s important to get a clear understanding of what Euclid means when 
he speaks of sne number multiplying another, vti. Def, 15 states that the 
effect of "a multiplying b" is taking a times b. We shall always represent 
" a times b " by ab and " b times a " by ba. This being premiseidj the proof 
that ab = ba may be represented as follows in the language of proportions. 



VII. i6, ij] PROPOSITIONS i6, 17 ^p 

. ... Ku 
[vii. 13] 



By V!i. Def. 10, 


I : a = i : ai. 


Therefore, alternately, 


I : t = a : a&. 


Again, by vii. Def. 10, 


X : i = a : ba. 


Therefore 


a : ab = a \ ba. 



or ttb-ba. 

Euclid does not use the language of proportions but that of fractions or 
their equivalent measures, quoting vn, 15, a particular case of vii, 13 
differently expressed, instead of vii, 13 itself. 

Proposition 17. 

If a number by muUiplymg itvo 7iuml>ers make certain 
numbers, the numbers so produced will have ike same ratio 
as the numbers multiplied. 

For let the number A by multiplying the two numbers B, 
C make D, B\ - ' ' 

I say that, as .5 is to C, so is D to E. 

For, since A by multiplying B has made D, 
therefore B measures D according to the units in A. 

A 



B C- 



F 

But the unit /^also measures the number A according to 
the units in it ; 

therefore the unit F measures the number A the same number 
of times that B measures D. 

Therefore, as the unit P is to the number ^, so is ^ to D. 

[vii. Def. 30] 

For the same reason, 
as the unit F is to the number A, so also is C to £■ ; 
therefore also, as .5 is to /?, so is C to £. 

Therefore, alternately, as B is to C, so is D to E. [vn. 13] 

Q. E. D. 
b K^ab \ ac. 
In this case Euclid translates the language of measures into that of 
proportions, and the proof is exactly like that set out in the last note. 
By VII. Def. so, i : a = b : ai, 

and X -.a-c: ac. 

Therefore b : ah^c \ac, 

and, altematdy, 6:c = ai:at. ' [*"• 'S] 



$1* BOOK VII ^« [vii. 18,19 



' " ' Proposition 18. ,5.ui-.!jiT 

// two numbers by multiplying any number make certain 
numbers, the numbers so produced will have the same ratio as 
the multipliers. 

For let two numbers A, B \yj multiplying any number C 
make D, E ; 
I say that, as A is to B, so is D c 

For, since A by multiplying e 

C has made Z?, 

therefore also C by multiplying A has made Z?. [vil 16] 

For the same reason also ^ ,. ^, 

C by multiplying B has made E. 

Therefore the number C by multiplying the two numbers 
A, B has made D, E. 

Therefore, as j4 is to B, so is D to E. [vii. 17] 

It is here proved that a:b=€ie:be. 

The argument is as follows. 

ac = ea. [vil, i6] 

Similarly 6( - cb. 

And a:i = ca:€bi ["I- 17] 

therefcne a : b = tu : 6e. 

• ' '! ■ .l.l-i;' 

■.-. I , 

Proposition 19. 

If four numbers be proportional, the number produced from 
the first and fourth will be equal to the number produced from 
the second and third; and, if the number produced from the 
first and fourth be equal to that produced from the second and 
third, the four numbers will be proportional. 

Let A, B, C, Dhe four numbers in proportion, so that, 
as j4 is to B, so is C to i? ; 
and let A by multiplying D make £, and let B by multiply- 
ing C make E; 
I say that E is equal to E. 

For let A by multiplying C make G, 



VII. 19] 



PROPOSITIONS 18, 19 



319 



Since, then, A by multiplying C has made G, and by 
multiplying D has made E, 
the number A by multiplying the two 
numbers C, D has made G, E. 

Therefore, as C is to Z?, so is G to E. 

[vu. 17] 

But, as C is to V, sols A 10 B ; 
therefore also, as ^ is to 5, so is C 
to E. 

Again, since A by multiplying C 
has made G, 

but, further, B has also by multiplying 
C made F, 

the two numbers A, B by multiplying a certain number C 
have made G, E. 

Therefore, as A is to B,so\s G to E. ' [vii. 18] 

But further, as A is to B, ^i is G to E also ; 
therefore also, as G Is to E, so is G to F. 

Therefore G has to each of the numbers E, E the same 
ratio ; 

therefore E is equal to E. 

Again, let E be equal to E; 
I say that, as .^ is to B, so is C to D. 

For, with the same construction, 
since E is equal to F, 
therefore, as (? is to E, so is G to F. 

But, as £7 is to E, so is C to D, 

and, as G is to E, so is A to B. 

Therefore also, as A is to ^, so is C to D. 

■ -' ' ' J. y. E. D. 

■ . r. I.. ■■ 

If ■•; I:. a :b = e:d, 

then ad=kc; and conversely. 
The proof is equivalent to the following, 

(i) at : ad—{ : d 

■f\ if.xhi •■. I I =a : i, '■•" 

Bui ^"^ ,W a:»^acii(. 

Therefore _• ^ i ■ • m : ad = ae : be, 

or ad=be. 



[cf. V. 9] 



^ ••• a y 




■•'l-.v . , 


[cf. V. 7] 




[vn. 17] 




[VII. 18] 



[vn. 17] 
'" • (vn. 18] 



3M BOOK VII [tu. 19, 10 





BOOK VII 


) Since 


ad = b<i. 




ac \ ad = ae ; be. 


But 


ac : ad- e : d. 


and 


ac;bc~a : b. 


Therefore 


a \b = c \ d. 



[vii. 17] 
[VII. iB] 

As indicated in the note on vii. 14 above, Heiberg regards Euclid t& 
basing the inferences contained in the last step of part (i) of this proof and 
in the first step of part (2) on the propositions v, 9 and v. 7 respectivefy, 
since he has not proved those propositions separately for numbers in this 
Book. I prefer to suppose that he regarded the inferences as obvious and 
not needing proof, in view of the definition of numbers which are in pro- 
portion. E.g., if at is the same fraction (" part " or " parts ") of ad that at is 
of be, it is obvious that ad must be ei^ual to be. 

Heiberg omits from his text here, and relegates to an Appendix, a 
proposition appearing in the manuscripts V, p, ^ to the effect that, if ihret 
numbers be proportional, the product of the extremes is equal to the square 
of the mean, and conversely. It does not appear in P in the first hand, B has 
it in the margin only, and Campanus omits it, remarking that Euclid does 
not give the proposition about three proportionals as he does in vi. 17, since 
it is easily proved by the proposition just given. Moreover an-Nairiri quotes 
the proposition about three proportionals at an obsemaiion on vii. 19 probably 
due to Heron (who is mentioned by name in the preceding paragraph). 



Proposition 20. _ 

The least numbers of those whitk have ike same ratio with 
them measure those which Jtave the same ratio the same number 
of times, the greater the greater and t lie less the less. 

For let CD, EF be the least niimbens of those which have 
the same ratio with A, B ; 
I say that CD measures A the same number 
of times that EF measures B. 

Now CD is not parts of ^. 

For, if possible, let it be so ; 
therefore EF is also the same parts of B 
that CD is of ^. [vn. 13 and Def. ao] 

Therefore, as many parts of A as there 
are in CD, so many parts of ^ are there also 
in EF, 

Let CD be divided into the parts of A, namely CG, GD, 
and £"^into the parts oi B, namely EH, HF\ 
thus the multitude of CG, GD will be equal to the multitude 
of EH, HF. 



o 



E 



VII. ao] PROPOSITIONS 19, 20 -3*1 

Now, since the numbers CG, GD are equal to one another, 
and the nunnbers EH, HF are also equal to one another, 

while the multitude of CG, GD is equal to the multitude of 
EH, HF, 

therefore, as CG is to EH, so is GD to HF, ' " " 

Therefore also, as one of the antecedents is to one of 
the consequents, so will all the antecedents be to all the 
consequents. [vu. u] 

Therefore, as CG is to EH, so is CD to EF. 

Therefore CG, EH are in the same ratio with CD, EF, 
being less than they : 

which is impossible, for by hypothesis CD, EF are the least 
numbers of those which have the same ratio with them. 

Therefore CD is not parts of A ; 

therefore it is a part of it. [vit, 4] 

And EF is the same part of B that CD is of ^ ; 

[vii. 13 and Def. 20] 
therefore CD measures A the same number of times that EF 
measures B. 

Q. E. D. 

1( a, b are the least numbers among those which have the same ratio 
(i.e. if ajb is a fraction in its lowest terms), and t, d are any others in the same 
ratio,, i.e. if 

a: b'^t :d, 

then o = - f and b = - d, where tt is some int^er. 

ft tt , 1 . _ I 

The proof is by reductio ad absurdum, thus. 

[Since a<c,a\% some proper fraction (" part " or "parts ") of e, l^ vii. 4.] 

Now a cannot be equal to —e, where m is an integer less than n but 

greater than i. 

For, if a = -f, b= -rfalso. [vii, 13 and Def. aol 

Take each of the m parts of a with each of the m parts of b, two and two ; 

the latio of the members of all pairs is the same ratio ~ a : ■- b. ^ 

mm 



Therefore 



- ix : - 6 = a:b. ' fvit. la] 

m nt 

But — a and — b are respectively less than a, b and they are in the same 
tit tti 

ratio : which contradicts the hypothesis. 



3» BOOK VII [vii. so, ji 

Hence a can only be " a part " of r, or 

a is of the form - c, 
ft 

and therefore d is of the form - rf. • , .•■ 

Here also Heibe^ omits a proposition which was no doubt interpolated 
by Theon (B, V, p, ^ have it as vii. 22, hut P only has it in the margin 
and in a later hand ; Campanus also omits it) proving for numbers the m 
aequali proposition when "the proportion is perturbed," i.e. (cf. enunciation 
of V. jj) if 

a:b = e\f, (i) 

and 6:e = d:e, (2) 

then a •.e = d:/. 

The proof (see Heiberg's Appendix) depends on vii. 19. 
From (i) we have of —be, 

and from (2) bt — cd. [vii. 19] 

Therefore af= cd, 

and accordingly a:c-d:f. ,. [vii. 19] 



Proposition 21. 

Numbers prime io one another are the least of those which 
have the same ratio with them, 

'■ Let A, B be numbers prime to one another; 
I say that A, B are the least of 
those which have the same ratio 
with them. 

For, if not, there will be some 
numbers less than A, B which are 
in the same ratio with A, B. 

Let them be C, D. 

Since, then, the least numbers of those which have the 
same ratio measure those which have the same ratio the 
same number of times, the greater the greater and the less 
the less, that is, the antecedent the antecedent and the 
consequent the consequent, [vii. 20] 

therefore C measures A the same number of times that D 
measures B. 

Now, as many times as C measures A, so many units let 
there be in E. 

Therefore D also measures B according to the units in E. 



C D 



I 



VII. ai, It] PROPOSITIONS 20— »a 313 

And, since C measures ^ according to the units in £, 
therefore £ also measures A according to the units in C. 

[vn. 16] 
For the same reason 

£ also measures B according to the units in D, [vii. 16] 

Therefore £ measures A, B which are prime to one 

another : which is impossible. [vn. Def. u] 

Therefore there will be no numbers less than A, B which 

are in the same ratio with A, B. 

Therefore A, B are the least of those which have the same 

ratio with them. 

Q. E. D. 

In other words, \\ a,b are prime to one another, the ratio a : J is " in its 
lowest terms." 

The proof is equivalent to the following. 

If not, suppose that f, </ are the least numbers for which 
a \b-e \ d. 
[Euclid only supposes tome numbers f, d in the ratio of o to i such that 
ir<a, and (consequendy) d-^b. It Js however necessary to suppose that 
f, d are the least numbers in that ratio in order to enable vn. 3o to be 
used m the proof.] 

Then [vn. ao] a = mt, and b = md, where m is some integer. 

Therefore o = cm, b - dm, [vn. 16] 

and m is a common measure of a, b, though these ate prime to one another . 
which is impossible. [vn. Def. la] 

Thus the least numbers in the ratio of <7 to ^ cannot be less than a, i 
thetnselves. 

Where I have quoted vn. i6 Heiberg regards the reference as being to 
VII. ■ 5. I think the phraseology of the text combined with that of Def. 15 
suggests the former rather than the latter. 

Proposition 22, 

T/te hast numbers of those which have the same ratio with 
them are prime to one another. 

Let .^, ^ be the least numbers of those which have the 
same ratio with them ; 

I say that A, B are prime to one g 

another. 

c 

For, if they are not prime to one d 

another, some number will measure ^ 

them. 

Let some number measure them, and let it be C 



^ BOOK VII [vii. aa, aj 

And, as many times as C measures A, so many units 
let there be in D, „:. -t rw-h 

and, as many times as C measures B, so many units let there 
be in E 

\<s Since C measures A according to the units in D, .<^w'ik 
therefore C by multiplying D has made A, [vii, Def. 15] 

For the same reason also 
C by multiplying E has made B. '• ' ■ 

Thus the number C by multiplying the two numbers /?, 
E has made A, B ; 

therefore, as D is to E, so is ^ x.o B\ [vii. 17] 

therefore D, E are in the same ratio with A, B, being less 
than they : which is impossible. 

Therefore no number will measure the numbers A, B. 

Therefore A, B are prime to one another. 

Q. E. U. 

\i a: b\%" in its lowest terms," a, b are prime to one another. 

Again the proof is indirect. 

If a, b are not prime to one another, they have some common measure f. 
Mid .„ . , 

a = m^, b = ne. 

Therefore m : n-a : b. [vii. 17 or iS] 

But m, n are less than a, b respectively, so that a ; A is not in its lowest 
terms : which is contrary to the hypothesis. 

Therefore etc. 



Proposition 23. .. . ., . > .. , -i-a,-''' 

If two numbers be prime to one another, (he number which 
measures the one of them wilt be prime to the remaining 
number. 

Let A, B be two numbers prime to one another, and let 
any number C measure A ; 
I say that C, B are also prime to one another. 

For, if C, B are not prime to one another, 
some number will measure C, B. 

Let a number measure them, and let it be D. 

Since D measures C, and C measures A, 
therefore D also measures A. a a c 6 

But it also measures B; 



vfi. 13, 2i\] PROPOSITIONS 2^—14 3»S 

therefore Z? measures A, B which are prime to one another : 
which is impossible, [vii, Def. la) 

Therefore no number will measure the numbers C B. 

Therefore C, B are prime to one another. 

Q. E. D. 

If a, mil are prime to one another, b is prime to a. For, if not, some 
number d will measure both a and *, and therefore both a and mb ; which is 
contrary to the hypothesis. 

Therefore etc. 

Proposition 24. 

If two numbers be prime to any number, their product also 

will be prime to the same. ' " " 

For let the two numbers A, B be prime to any number C, 
and let A by multiplying B make D \ 
I say that C, D are prime to one another. 

For, if C, D are not prime to one another, 
some number will measure C, D. 

Let a number measure them, and let it 
be^. 

Now, since C, A are prime to one 
another, 

and a certain number E measures C, 

therefore A, £ 3.re prime to one another. [v:t. 23] 

As many times, then, as E measures Z?, so many units let 
there hem E; 

therefore E also measures £> according to the units in E. 

[vii. 16] 

Therefore E by multiplying E has made 2?, [vit. Def. 15] 

But, further, A by multiplying B has also made D ; 
therefore the product of E, E is equal to the product of A, B. 

But, if the product of the extremes be equal to that of the 
means, the four numbers are proportional ; [vn. 19] 

therefore, as £ is to A, so is B to E. '* ^'' 

But A, £ are prime to one another, 
numbers which are prime to one another are also the least of 
those which have the same ratio, [vn. 21J 

and the least numbers of those which have the same ratio 
with them measure those which have the same ratio the same 



jalL BOOK VII [vii. n, a. 5 

number of times, the greater the greater iind the less the less, 
that is, the antecedent the antecedent and the consequent the 
consequent ; [vn. ao] 

therefore £ measures B. 

But it also measures C ; 
therefore £ measures B, C which are prime to one another : 
which is impossible. [vn. Def. i*] 

Therefore no number will measure the numbers C, D. 

Therefore C D are prime to one another. 

Q. E, D. 

I. their product. A ii viitSir ivtbpunt, literal); " the (number) pradoced rrom ihem," 
will liencefart)) be translated as "tjieir product." 

If n, b a^ both prime to c, then ab, c are prime to one another. 
The proof is again by reduciis ad absurdum. 

If ab, c are not prime to one another, let them be measured by a ^nd be 
equal to md, mi, say, respectively. 

Now, since a, c are prime to one another and d measures e, 

a, d are prime to one anbther. [vn. 33] 

But, since ab = md, 

d;a = b:m. [vn. 19] 

Therefore [vit. *o] d m^sures *, 

or b =pd, say. . 1 ,. 

But e = ttd. 

Therefore d measures both i and c, which are therefore not prime to one 
another : which is impossible. 

Therefore etc ■ '■ ' •'■ "" ' ' '" ^;!j 

Proposition 25. 

// two numbers bt prime to one another, the product of one 
of them into itself will be prime to the remaining one. 

Let A, B be two numbers prime to one another, . ., 
and let A by multiplying itself make C: „, „ j 

I say that B, C are prime to one another. 

For let D be made equal to A. 

Since A, B are prime to one another, 
and A is equal to D, 
therefore D, B are also prime to one another. 

Therefore each of the two numbers D, A is 
prime to B ; 
therefore the product of D, A will also be prime to B. [vn. 34] 



vii, IS, 16] PROPOSITIONS 24—26 ' J»J 

But the number which is the product of D, A is C 
Therefore C, B are prime to one another. q. e. d. 

I. the product of one of them into Itself. The Greeks h in raO hfht a6r«iw ytr&furott 
literatlj ^'the number produced from the one of them/' leaves '* multiplied into it$«lf '^ to be 
understood. 

If a, i are prime to one another, - v ■ ■ .' >' ■ ' • 

a' is prime to i. v 1 - 1 

Euclid takes d equal to a, so that d, a are both prime to ^. 

Hence, by vii. 24, da, i.e. a', is prime to i. 

The proposition is a particular case of the preceding proposition ; and the 
method of proof is by substitution of different numbers in the result of that 
proposition. 

' Proposition 26. 

If two numbers ie prime to two numbers, both to each, their- 
products also will be prime to one another. 

For let the two numbers A, B he prime to the two 
numbers C, D; both to each, 

and let A by multiplying B ^^ q 

make E, and let C by multi- g t, 

plying Z* make ./^; 

I say that E, F are prime to p 

one another. 

For, since each of the numbers A, B is prime to C, 
therefore the product oi A, B will also be prime to C. [vn. *4] 

But the product of A, B is E ; > 

therefore B, C are prime to one another. , , 

For the same reason ^, - m..-; /. 

Et D are also prime to one another. •., . .-n- u 

Therefore each of the numbers C, D is prime to E. 

Therefore the product of C, D will also be prime to E. 

[vii. *4] 

But the product of C, D is F. 

Therefore E, Fa.Tc prime to one another. q. e. d. 

If both a and i are prime to each of two numbers e, d, then ai, cd will be 
prime to one another. 

Since a, b are both prime to c, 

ab, t are prime to one another. [vn. 34] 

Similarly ab, d are prime to one another. 

Therefore c. d are both prime to ab, 

and so theiefoK is ed- [vn. 34] 



j(*8 



BOOK VII 



[vii. 27 



Proposition 27. 

If two numbers be prime to one another, and each by 
multiplying itself make a certain number, the products V}iU be 
prime to one another; and, if the original numbers by multi- 
plying the products make certain numbers, the latter will also 
be prime to one another [and this is always the case with the 
extremes\ 

Let A, B be two numbers prime to one another, 
let A by multiplying itself make C, and by 
multiplying C make D, 
and let B by multiplying itself make E, and 
by multiplying E make F; 
I say that both C, E and D, F are prime 
to one another. 

For, since A, B are prime to one another, 
and v4 by multiplying itself has made C, 
therefore C, B are prime to one another. [vn. 25] 

Since then C, B are prime to one another, 
and B by multiplying itself has made B, 
therefore C, E are prime to one another. [«£} 

Again, since A, B are prime to one another, 
and B by multiplying itself has made E, 
therefore A, E are prime to one another. \id:\ 

Since then the two numbers A, C are prime to the two 
numbers S, E, both to each, 

therefore also the product of A, C is prime to the product of 
B, E. [vn. 26] 

And the product of A, C is D, and the product of B, E 
is F. 

Therefore D, F are prime to one another. 

Q. E. D. 

If a, b are prime to one another, so are a', ^ and so are a*, ^; and, 
generally, a", i" are prime to one another. 

The words in the enunciation which assert the truth of the proposition for 
any powers are suspected and bracketed by Heiberg because (i) in jripl rots 
Bitpout the use of ojtpoi is peculiar, for it can only mean " the last products," 
and _(*) the words have nothing corresponding to them in the proof, much 
less ts the generalisation proved. Campanus omits the words tn the enuncia'^ 



vii. *7, a8] PROPOSITIONS 37, 18 3*9 

tion, though he adds to tht proof a remark that the proposition is true of any, 
the same or different, powers of «, i. Heiberg concludes that the words are 
an interpolation of date earlier than Theon. 

Euclid's proof amounts to this. 

Since a, i are prime to one another, so are a*, * [vii. a si. and therefore 
also a', ff. [vii, as] 

Similarly [vii. as] «, ^ are prime to one another. 

Therefore a, a* and b, *■ satisfy the description in the enunciation of 
VII. a 6. 

Hence a', ^ are prime to one another. .■ . • r > ^ > ...,..,,, 



Proposition 28. 

If two numbers be prime to one another, the sum will also 
be prime to each of tkem ; and, if the sum of two numbers be 
prime to any one of them, ike original numbers will also be 
prime to one another. 

For let two numbers AB, BC prime to one another be 
added ; 

I say that the sum AC \s also prime a" ■ ~S 6 

to each of the numbers AB, BC. 

D 

For, if CA, AB are not prime to .^ , 

one another, 

some number will measure CA, AB. 

Let a number measure them, and let it be Z). 
Since then D measures CA, AB, 

therefore it will also measure the remainder BC. ,^ ^^ | 

But it also measures BA ; 

therefore D measures AB, BC which are prime to one another : 
which is impossible. [vii, Def. u] 

Therefore no number will measure the numbers CA, AB; 
therefore CA, AB are prime to one another. 
For the same reason 

AC, CB are also prime to one another. 

Therefore CA is prime to each of the numbers AB, BC. 

Again, let CA, AB he prime to one another ; 
I say that AB, BC are also prime to one another. 

For, if AB, BC are not prime to one another, 
some number will measure AB, BC. 



330 BOOK VII [vn. zS, 19 

Let a number measure them, and let it be />. 

Now, since Z? measures each of the numbers AB, BC, it 
will also measure the whole CA. 

But it also measures AB ; 
therefore D measures CA, AB which are prime to one another: 
which is impossible. [vii. Def. la] 

Therefore no number will measure the numbers AB, BC. 

Therefore AB, BC are prime to one another. 

Q. E. D. 

If a, b are prime to one another, a-¥6 will be prinm to both a and b ; and 
conversely. 

For supjwse (a + *), a are not prime to one another. They must then 
have some common measure d. 

Therefore d also divides the difference {a + li) - a, or b, as well as a ; and 
therefore a, h are not prime to one another : which is contrary to the 
hypothesis. 

Therefore a + i is prime to a. 

Similarly b + * is prime to #. , , , 1, 

The converse is proved in the same way. 

Heibei^ remarks on Euclid's assumption that, if c measures both a and b, 
it also measures a±b. But it has already (vji. i, j) been assumed, more 
generally, as an axiom that, tn the case supposed, c measures a±J>b. 



Proposition 29. : ..j - . 1 > 

Any prime number is prime to any number which it does 
not measure. 

Let ^ be a prime number, and let it not measure B ; 
I say that B, A are prime to one another. 

For, if B, A are not prime to one ^a 

another, — ^b 

some number will measure them. c 

Let C measure them. 

Since C measures jff, 
and A does not measure B, 
therefore C is not the same with A. 

Now, since C measures B, A, 
therefore it also measures A which is prime, though it is not 
the same with it : 



which is impossible. 



vn. S9, 30) PROPOSITIONS iS— .10 $$i 

Therefore no number will measure B, A. 
Therefore A, B are prime to one another. 

Q. E. D. 

If a is prime and does not mtsasure b, tht:n a, b axa priiiiu lo onu anuthur. 
The proof is self-evident. 



Proposition 30. 

If two numbers by muliiplying one another utake some 
number, and any prime number measure the product, it will 
also measure one of the original numbers. 

For let the two numbers A, B by multlplvirig one another 
make C, and let any prime number 

D measure C ; ^ , 

I say that D measures one of the b_ 

numbers A, B. c ■ 

For let it not measure A. x) 

Now D is prime ; e 

therefore A, D are prime to one 

another. [vu. 29J 

And, as many times as D measures C, so many units let 
there be in .£". 

Since then D measures C according to the units in E, 
therefore D by multiplying E has made C. [vk. IJef, 15] 

Further, A by multiplying B has also made C\ 
therefore the product of D, E is equal to the product of 
A, B. 

Therefore, as /? is to ^, so is .5 to E. [vu, 19] 

But D, A are prime to one another, 
primes are also least, [vu. zi] 

and the least measure the numbers which have the same 
ratio the same number of times, the greater the greater and 
the less the less, that is, the antecedent the antecedent and 
the consequent the consequent ; , -, [vu. 20] 

therefore D measures B. 

Similarly we can also show that, if D do not measure B, 
it will measure A. 

Therefore D measures one of the numbers A, B. 

Q. E. u. 



332 '.'ir BOOK VII [vii. 30, 31 

If IT, a primt; number, measure ai, e will measure either a ur i. 
Suppose c does not measure a. 

Therefore ^, a are prime to one another. [vii, 19] 

Suppcrae ab-me. 

Therefore e:a-b:m. .- 1; ^ ■' . 'i.i. >'< [vu. 19] 

Hence [vii. JO, 2i] fmeasuresi, ' •■-'• " 

Similarly, if c does not measure l>, it measures c. 
Therefore it measures one or other of the two numbers a, k 



'■.■• •t'' 'i S PkOVOHITION 31. • • ^•''• 

Any compost ie number is measured by some prime number. 

Let ^ be a composite number ; . . ' 

I say that A is measured by some prime number. 

For, since A is composite, ' ' 
S some number will measure it A 

Let a number measure it, and let it b- — 

be B. c— 

Now, if B is prime, what was en- 
joined will have been done. 
10 But if it is composite, some number will measure it 

Let a number measure it, and let it be C 

Then, since C measures B, 
and .5 measures .^, ■•:-,••• .■•''■ • • ■ ' 

therefore C also measures >(4. ■■ 

IS And, if C is prime, what was enjoined will have been 
done. 

But if it is composite, some number will measure it 
Thus, if the investigation be continued in this way, some 
prime number will be found which will measure the number 
a) before it, which will also measure A. 

For, if it is not found, an infinite series of number will 
measure the number A, each of which is less than the other; 

which is impossible in numbers. 

Therefore some prime number will be found which will 
as measure the one before it, which will also measure A. 

Therefore any composite number is im^asured by some 
prime number. 



V". 3>— 33] PROPOSITIONS 30—33 333 

S. if B Is prime, what was enjoined vrlll have been done, i.e. the implied 
prsiiim of finding n prime number which measures A* 

iS. some prime number will be found which will measure. In the Creek the 
sentence stops here, but it U necessary to add the words " the number before it, which will 
also measure j4," which are found a few lines further down. It is possible that the words 
ma^ have fidlen out of P here by a simple mistake due to biitnari\ivtae (Heiberg). 

Heiberg relegates to the Appendix an alternative proof of this proposition, 
to the following effect. Since A is composite, some number will measure it. 
Let B be the least such number. I say that 3 is prime. For, if not, B is 
composite, and some number will measure it, say C; so that C is less than B. 
But, since C measures S, and B tneasures A, C must measure A. And C is 
less than B -. which is contrary to the hypothesis, 



• '• " Proposition 32. '' "' ' ■ 

Any number either is prime or is measured by some prime 

number, 

• ■. .-. t-j.i .V 1-. . • • 

Let .^4 be a number; 
I say that A either is prime or is measured by some prime 
number. , , 

If now A IS prime, that which was a '■ 

enjoined will have been done. 

But if it is composite, some prime number will measure it 

[vii. 31] 

Therefore any number either is prime or is measured by 
some prime number. 

Q, E. D, 

Proposition 33, 

Gwen as many numbers as we p/ease, to find the least of 
those which have the same ratio with them. 

Let A, B, C be the given numbers, as many as we please ; 
thus it is required to find the least of 
s those which have the same ratio with 
A, B, C. * 

A, B, C are either prime to one 
another or not. 

Now, \i A, B, C are prime to one I I 
10 another, they are the least of those y^ I 
which have the same ratio with them, 

[vii. 3i] 

But, if not, let D the g^-eatest common measure of .^, B, C 
be taken, [vii. 3] 



B 


C 




E 




1- t 


1 


1 


Q 



334 BOOK VII IDJtn , [VI,. 33 

and, as many times as D measures the numbers A, B, C 
IS respectively, so many units let there be in the numbers 
£, F, G respectively. 

Therefore the numbers E, F, G measure the numbers A, 
B, C respectively according to the units in D. \yu. 16] 

Therefore E, E, G measure A, B, C the same number of 
30 times ; 

therefore E, E, G are in the same ratio with A, B, C. 

[vii. Drf. 30] 

I say next that they are the least that are in that ratio. 
For, if E, E, G are not the least of those which have the 
same ratio with A, B, C, 

'S there will be numbers less than E, F, G which are in the 

same ratio with A, B, C. 

Let them )x. H, K, L ; .-..-. 

therefore H measures A the same number of times that the 

numbers K, L measure the numbers B, C respectively. 
30 Now, as many times as H measures A, so many units let 

there be in M; 

therefore the numbers K, L also measure the numbers B, C 
r^pectively according to the units in M. 

And, since H measures A according to the units in M, 

35 therefore M also measures A according to the units in H. 

[vn. 16] 
For the same reason 

M also measures the numbers B, C according to the units in 
the numbers K, L respectively ; 

; - Therefore M measures A, B, C. 
40 Now, since H measures A according to the units in M, 
therefore H by multiplying M has made A. [vn. Def. 15] 

For the same reason also ,i •. 

E by multiplying D has made A. 

Therefore the product oi E, D\& equal to the product of 
4S H,M. 

Therefore, as ^ is to H, so is M to D. [vn. 19] 

But E is greater than H ; 

therefore M is also greater than D, 
It And it measures A, B, C\ 



VII. 33j PROPOSITION 33 38S 

JO which is impossible, for by hypothesis D is the greatest 
common measure of A, B, C. 

Therefore there cannot be any numbers less than E, F, G 
which are in the same ratio with A, B, C. 

Therefore F, F, G are the least of those which have the 
js same ratio with A, B, C. 

•rviT i.iii'w •i^,\tTr.m nefH ".n* ! r-;( uJ ^^jwt.,- Q- E. D. 

17. the numbeis E, F, C measure the numbers A, B, C respectively, 
literally (as usual) "eacli of the numl>er» £, F, G measures each of the numbers A, 



— rr 



Given any numbets a, h, c, .,,, to find the least numbers that are in the 
same ratio. 

Euclid's method is the obvious one, and the result is verified by reductio 
ad absurdum. 

We wi!t, like Euclid, take three numbers only, a, h, c. 
Letf, their greatest common measure, be found [vn. 3], and suppose that 
a = mg, i.e. gm, , v> . . [vn. 16] 

6 = ug, U.g«. ,j^^ ,. 

': = />i. I.e. ^. X , , •.•.„■-; 

It follows, by vn. Def. 20, that ' • v 1 . 

4 1 ^ t .-* n :[w 

H m : H ■.p = a : i :e, j . . i 

«, », / shall be the numbers required. 

For, if not, let x, y, e be the least numbers in the same ratio as a, b, r, 
being less than /«, n, p. 

Therefore a = kx (or xk, vn. 16), 

i> = ky {at yk), \ i,l -^S 

c = kz (or zk), • 

where k is some integer. [vii. ao] 

Thus «lj{*= a = xk. 

Therefore "'^■'i ' '' "• m:x = k:g. .fj rj'.! [vii. 19] 

And m^ x; therefore k-> g. 

Since then k measures a, A, c, it follows that g is not the greatest comaon 
measure ; which contradicts the hypothesis. 

Therefore etc. "^ ^■'- ^-ifr-'I'T 

It is to be observed thai Euclid merely supposes that x, y, g are smaller 
numbers than m, n, p in the ratio of a^b, c\ but, in order to justify the next 
inference, which apparently can only depencl on vn. ao, x, y, t must also be 
assumed to be the hait numbers in the ratio of «, b, c. 

The inference from the last proportion that, since m> x, i >^is supposed 
by Hetberg to depend upon vit. 13 and v. 14 together. I prefer to regard 
Euclid as making the inference quite independently of Book v. E.g., the 
proportion could just as well be written 

X : m=g : k, ''^^ 

when the definition of proportion in Book vn. (Def. 20) gives all that we want, 
since, whatever proper fraction x is of m, the same proper fraction is g of k. 



^ BOOK VII [vii. 34 



Proposition 34. 

Given two numbers , to find the least number which th^ 
measure. 

Let A, B be the two given numbers ; 
thus it is required to find the least number which they 
measure, 

Now^,^ are either prime to one ^ B 

another or not. ^ 

First, let A, B be prime to one ^ 

another, and let A by multiplying B 

make C; ^ ^ 



therefore also B by multiplying A has . 

made C, : . • [vil 16] 

Therefore A, B measure C 

I say next that it is also the least number they measure. 

For, if not, A, B will measure some number which is less 
than C 

Let them measure D. , „ . 

Then, as many times as ^ measures D, so many units let 
there be in £, 

and, as many times as B measures D, so many units let there 
be in f; 

therefore A by multiplying £ has made D, 

and B by multiplying /^ has made Z? ; [vii. Def, 15] 

therefore the product of ^, j6" is equal to the product of ,5, F. 

Therefore, as A is to B, so is /^ to £. [vn, 19] 

But A, B are prime, 
primes are also least, [vn, ti] 

and the least measure the numbers which have the same ratio 
the same number of times, the greater the greater and the less 
the less ; [vn. 10] 

therefore B measures B, as consequent consequent. 

And, since A by multiplying B, E has made C. D, 
therefore, as B is to E, so is C to Z>. [vn. 17] 

But B measures E \ 
therefore C also measures D, the greater the less : 
which is impossible. 



Til. 34] PROPOSITION 34 337 

Therefore A, B do not measure any number less than C ; 
therefore C is the least that is measured by A, B. 

Next, let ^, B no*" be prime to one another, 
and let F, E, the least numbers of those which have the same 
ratio with A, B, be taken ; [vil 33] 

therefore the product of -^, ^ is equal to the product of ^, F. 

[vii. 19] 

And let A by multiplying E "" m,i«7>- , 

make C ; * b 

therefore also B by multiplying F p e 

has made C ; 

therefore A, B measure C. d •' ,, 

I say next that it is also the least h 

number that they measure. 

For, if not. A, B will measure some number which is less 
than C. 

Let them measure D. 

And, as many times as A measures D, so many units let 
there be in G, 

and, as many times as B measures D, so many units let there 
be in H. 

Therefore A by multiplying G has made D, 
and B by multiplying H has made D. 

Therefore the product of A, G is equal to the product of 
B,H; 
therefore, as A is to B, so is H to G. [vil 19] 

I, But, as y4 is to B, so is F to E. ,^^i, ..■ 

Therefore also, as /^ is to £, so is li to G. ^,^^ 

But F, E are least, ,j_^, , _ 

and the least measure the numbers which have the same ratio 

the same number of times, the greater the greater and the 

less the less ; ^ ^ [vn. to] 

therefore E measures G. 

And, since A by multiplying E, G has made C, /?, 

therefore, as £ Is to G, so is C to D. [vn. 17] 

But £ measures G ; itj v .•»». 

therefore C also measures D, the greater the less : 

which is impossible. 



^ BOOK VII [vii. 34 

Therefore A, B will not measure any number which is less 
than C. 

Therefore C is the least that is measured by A, B. 

Q, E. D. 

This is the problem of finding the itait common multipk of two numbers, 

as a, b. .. , . 

I, If «, ^ be prime to one another, the l.cm. is ab, , 

For, if not, let it be rf, some number less than aA 
Then d-ma-nb, where wi, n are int^ers. 

Therefore a.b-n:m, [vil. rg] 

and hence, a, b being prime to one another, 

b measures m. [vii. ao, »i} 

But b\M = iA;aM [vti. 17] 

= ab:d. 
Therefore ah measures d: which is impossible. 

' TT. If a, i be not prime to one another, find the numbers which are the 
least of those having the ratio of a to b, say «, « ; [vit. 33] 

then a: b^m -.n, 

and an-bm (=f, say); [vir, 19] 

e is then the i.c.m. 

For, if not, let it be i/ (< c), so that 

ap-bq = d, where/, q are integers. 

Then a:6 = q:p, [vii. 19] 

whence m : n = q -.p, 

so that n measures/. [vii. 20, ai] 

And n : p = an • ap = c : d, 

90 that e measures d: 

which is impossible. ' " 

Therefore etc. 



By VII 


33> 




m 
n 


4 

~~S 

_b 
^ s 


Hence the 


I.C.M 


. is 


ab 

S 



, where £ is the g.c.m. of a, b. 



vii. 35> 36] propositions 34—36 339 

Proposition 35. 

// two numbers measure any number, ike least number 
measured by tkem will also measure the same. 

For let the two numbers A, B measure any number CD, 

and let E be the least that they 

measure; * p 

I say that E also measures CD. 

For, if E does not measure 
CD, let E, measuring DF, leave C/^less than itself. 

Now, since A, B measure E, •'• 

and E measures DF, 
therefore A, B will also measure DF. 

But they also measure the whole CD ; 
therefore they will also measure the remainder CF which is 
less than E: ^ 

which is impossible. - '. 

Therefore E cannot fail to measure CD ; ,-i- ,.., -.j 
therefore it measures it. 

Q. E. D. 

The koii common multiple of any two numbers must measure any other 
common multiple. 

The proof is obvious, depending on the fact that, if any number divides » 
and b, it also divides a -ph. _i n. ; 



Proposition 36, 

Given three numbers, to find the least number which they 
measure. 

Let A, B, C be the three given numbers ; 
thus it is required to find the least . , 

number which they measure, a '"' .' 

Let D, the least number mea- b 

sured by the two numbers A, B, c — 

be taken. [vn. 34] d- 



Then C either measures, or e I'l 

does not measure, D. , . •_, 

First, let it measure it. 



.$4a • BOOK VII [VII. 36 

But A, B also measure D\ ■■■ . .-- - . - — t 

therefore A, B, C measure D. 

I say next that it is also the least that they measure. 

For, if not, A, B, C will measure some number which is 
less than D. 

Let them measure E. 

Since A, B, C measure E, ' 
therefore also A, B measure E. 

Therefore the lease number measured by A, B will also 
measure E. [vn. 35] 

But D is the least number measured by ^, Z? ; 
therefore D will measure E, the greater the less : 
which is impossible. 

Therefore A, B, C will not measure any number which is 
less than D ; 

therefore D is the least that A, B, C measure. 

Again, let C not measure D, ' 

and let E, the least number measured by . 

C, D, be taken. [vn. 34] 

Since A, B measure D, q 

and D measures E, ' 

therefore also A, B m^sure E, 1 

But C also measures E ; ' - ' ^ 

therefore also A, B, C measure E, 

1 say next that it is also the least that they measure. 
For, if not. A, B, C will measure some number which 
is less than E. 

Let them measure E. _;.rs 

Since A, B, C measure E, 

therefore also ^, ^ measure /^; 

therefore the least number measured by A, B will also 
measure E. [vn. 35] 

But D is the least number measured by A,*B ; 
therefore D measures F. , .,v •-; 

But C also measures /^; ejiii-^i.v ..'•'. 

therefore A C measure ./% 
so that the least number measured by D, C will also measure E. 



VII- 3<S, 37] PROPOSITIONS 36. 37 3*^ 

But £ is the least number measured by C, D; 
therefore £ measures J^, the greater the less : 
which is impossible. 

Therefore A, B, C will not measure any number which is 
less than £. 

Therefore £ is the least that is measured by A, B, C 

Q. K. D. 

Euclid's rule for finding the ucm. of ihrte numbers a, j, ^ is the rule with 
which we are familiar. The L.CI1. of a, b is first found, say d, and then the 
L.C.H. of d and c is found. 

Euclid distinguishes the cases (i) in which c measures d, (i) in which c 
does not measure d. We need only reproduce the proof of the general case 
(3). The method is that of rtducfio ad a&surdum. 

Let e be the L.C.M, of d, c. 

Since a, i both measure d, And d measures «, '' ' "^ ' 

a, b both measure e. 

So does c. 

Therefore e is iome common multiple of a, b, c. 

If it is not the ioj^, let/be the L.CM. 

Now a, b both measure/; 
therefore d, their L.aM., also measures/ [vu, 35] 

Thus d, ( both measure/ 
therefore e, their l.c.m,, measures/: : ,, . •• [vii. 35] 

which is impossible, since /< e. 

Therefore etc. 

The process can be continued ad libitum, so that we can find the L.C.M., 
liot only of threes but of as many numbers as we please. 



Proposition 37. ' '■ 

If a number be measured by any number, the number which 
is measured will have a part called by the same name as the 
measuring number. 

For let the number A be measured by any number B \ 
I say that A has a part called by the same 
name as B, a 

For, as many times as B measures A, b 

so many units let there be in C. ^ 

Since B measures A according to the p 

units in C, 

and the unit D also measures the number C according to the 

units in it, 



■^ XI / BOOK VII m [VII. 37, gS 

therefore the unit D measures the number Cthe same number 
of times as B measures A. 

Therefore, alternately, the unit D measures the number B 
the same number of times as C measures A ; [vii. 15] 

therefore, whatever part the unit D is of the number B, the 
same part is C of -^ also. 

But the unit D is a. part of the number B called by the 
same name as it ; 

therefore C is also a part of A called by the same name as B, 
so that A has a part C which is called by the same name as B. 

Q. E. D. 

If 6 measures a, then 7 th of ii is a whole number. 

Let a-m.b. 

Now m = m.x. 

Thus It m, b, a satisfy the enunciation of vti. 15 ; 
therefore m measures a the same number of times that i measures b. 



But I is T th part of b ; '' 

theiefore w is r th part of a. 



1 .- 'i^, 



Proposition 38. ;r. 

// a number have any part whatever, it will be measured 
by a number called by the same name as the part. 

For let the number A have any part whatever, B, 
and let C be a number called by the same 
name as the part B ; 
I say that C measures A. * ' 

For, since ^ is a part of A called by ^ 

the same name as C, ^ 

and the unit D is also a part of C called 
by the same name as it, 

therefore, whatever part the unit D is of the number C, 

the same part \s B o( A also ; 

therefore the unit D measures the number C the same number 
of times that B measures A. 



viL 38, 39j PROPOSITIONS 37—39 343 

• Therefore, alternately, the unit D measures the number B 
the same number of times that C measures A. [vit. 15] 

Therefore C measures A. 

n I . J >i/i 3d' Q. E. D. 

This proposition is practically a. restatement of the preceding proposition. 
It asserts that, if * is - th part of «, ,._,., 

i.e.,if <5 = -a, . J -^ ^ '-'i ! 

m . . , 

then m measures a, 

.131-11, J i? « '' i I' .ri A 

We have . .; . b=-~ a, . •, 



■1- 



m 



and !=—«(. 

Therefore i, m, i, a, satisfy the enunciation of vit. 15, and thet«foTe m 
measures n the same number of times as i measures i, or 

I 

' , n( = 1 a. . ._. . . 1- 

r. i ■ - '" ' '•• " 



Proposition 39. ill 

To find the number which is the least that will have given 
parts. 

Let A, B, C be the given parts ; 
thus it is required to find the number which is the least thai 
will have the parts A, B, C. 

A B c, 

D 



Let D, £, F be numbers called by the same name as the 
parts A, B, C, 

and let G, the least number measured by D, E, 7^ be taken. 

[vii. 36] 

Therefore G has parts called by the same name as D, E, F. 

[vn. 37] 
But A, B, C are parts called by the same name as Z>, E, F\ 
therefore G has the parts A, 3, C. 

I say next that it is also the least number that has. 



344 BOOK VII [viL 39 

For, if not, there will be some number less than G which 
will have the parts A, B, C. 

Let it be H. m - :n' * 

Since H has the parts A, By C, 
therefore H will be measured by numbers called by the same 
name as the parts A, B, C. fvu. 38] 

But D, E, F are numbers called by the same name as the 
parts A, B, C\ 
therefore H is measured by D, E, F. 

And it is less than G : which is impossible. 

Therefore there will be no number less than G that will 
have the parts A, B, C. 

Q. E. D, 

This again is practically a restatement in another form of the problem of 
finding the L.C.M. 

To find a number which has - th, t th and - th parts. 

Let d be the l.c.m. of a, d, c. 

Thus d has -eh, rth and -th parts. [vti. 37] 

If it is not the least number which has, let the least such number be <■. 

Then, since e has those parts, 
e is measured by a, 6,e; and e<d: 
which is impossible. 






BOOK VIII. 



Proposition i. 

■r»f 

// there be as many numbers as we please in continued 
proportion, and the extremes of them be prime to one another, 
the numbers are the least of those which have the same ratio 
with them. 

Let there be as many numbers as wk please, A^ B, C, D, 
in continued proportion, 

and let the extremes of them a- ~ g — 

A, D\x. prime to one another; b f 

I say that A, B,C, D are the Q 

least of those which have the " h 

same ratio with them. 

For, if not, let E, F, G, H h^ less than A, B, C, D, and 
in the same ratio with them. 

Now, since A, B, C, D are in the same ratio with E, F. 
G,H, 

and the multitude of the numbers A, B, C, D is equal to the 
multitude of the numbers E, F, G, H, 
therefore, ex aequali, 

as A is to D, so is E to H. [vii. 14] 

But A, D are prime, 
primes are also least, [vii. ai] 

and the least numbers measure those which have the same 
ratio the same number of times, the greater the greater and 
the less the less, that is, the antecedent the antecedent and 
the consequent the consequent. [vn. ao] 



346 BOOK Vin [vni. i, i 

Therefore W measures £, the greater the less : 
which is impossible. 

Therefore E, F, G, H which are Jess than A^ B, C, D 
are not in the same ratio with them. 

Therefore A, B, C, D are the least of those which have 
the same ratio with them. 

Q. E. D. 

What we call a geometrical progression is with Euclid a series of terms "in 
continued proportion " (ifijt o'lutAo^}. 

This proposition proves that, if a, f, <:,... ji are a series of numbers in 
geometrical progression, and if o, k are prime to one another, the series is in 
the lowest terms jxj^ible with the same common ratio. 

The proof is in femi by redudio ad absurdum. We should no doubt 
desert \.\\\&form while retaining the substance. If «', iJ', c', . . . A' be any other 
series of numbers in c.p. with the same common ratio as before, we have, 
tx atquali, 

a : k = a' : k', [vii. 14] 

whence, since a, k are prime to one another, a, k measure a', k' respectively, so 
that a', k' are greater than a, k respectively. 

Proposition 2. 

To find numbers in continit^d proportion, as many as may 
be prescribed, and Ike least that are in a given ratio. 

Let the ratio of ^4 to ^ be the given ratio in least 
numbers ; 

thus it is required to find numbers in continued proportion, 
as many as may be prescribed, and the least that are in the 
ratio of A to j9. 



-^0 



Let four be prescribed ; 
let A by multiplying itself make C, and by multiplying B let 
it make D \ 

let B by multiplying itself make E ; 
further, let A by multiplying C, D, E make F, G, H, 
and let B by multiplying E make K. 



VIII. 2] PROPOSITIONS 1,1 ^j 

Now, since A by multiplying itself has made C, " *' 

and by multiplying B has made /?, 
therefore, as A is to B, so is C to /?. [vii. 17] 

Again, since A by multiplying B has made D, 

and ^ by multiplying itself has made £, 

therefore the numbers A, B hy multiplying £ have made the 
numbers D, E respectively. 

• 'r'l 

Therefore, as ^ is to B, so is D to E. [vii. 18] 

But, as y4 is to .5, so is C to Z? ; 
therefore also, as C is to D, so is D to E. 

And, since A by multiplying C, D has made F, G, 
therefore, as C is to D, so is F to C [vn. 17] 

But, as C is to D, so was A to B\ 
therefore also, as ^ is to B, so is F to G, 

Again, since A by multiplying /?, ^ has made G, //, 
therefore, as ZJ is to E, so is G to //'. [vii, 17] 

But, as i? is to £", so is /4 to B. 

Therefore also, as A is to B, so is G to H. 

And, since .^4, ^ by multiplying E have made H, K, 
therefore, as A is to B, so is H to ^. [vii, 18] 

But, as A is to B, so is /^ to G, and (7 to jfiT. 

Therefore also, as F is to G, so is G to H, and H io K; 
therefore C, Z?, £", and Z^ G, H, K are proportional in the 
ratio of A to B. 

I say next that they are the least numbers that are so, 

For, since A, B are the least of those which have the 
same ratio with them, 

and the least of those which have the same ratio are prime 
to one another, , ,,1 [vn. aa] 

therefore A, B are prime to one another. 

And the numbers A, B hy multiplying themselves re- 
spectively have made the numbers C, E, and by multiplying 
the numbers C, E respectively have made the numbers F, K\ 
therefore C, E and F, A'are prime to one another respectively, 

[vn. 27] 

But, if there be as many numbers as we please in continued 
proportion, and the extremes of them be prime to one another. 



348 BOOK VIII [vin. a, 3 

they are the least of those which have the same ratio with 
them. [viii. i] 

Therefore C, D, E and F, G, H, K are the least of those 
which have the same ratio with A, B. q. e. d. 

PokisM. From this it is manifest that, if three numbers 
in continued proportion be the least of those which have the 
same ratio with them, the extremes of them are squares, and, 
if four numbers, cubes. 

To find a series of numbers in geometrical progression and In the least 
terms which have a given common ratio (understanding by that term ilu ratio 
of one term to the next). 

Reduce the given i^tio to its lowest terms, say, a : i. (This can be done 
by VII. 33.) 

Then a", a*-'i, a'-^lr', ... a'*"-', u^-', i- 

is the required series of numbers if {« + i ) terms are required. 

That this is a series of terms with the given common ratio is clear from 
vit. 17, i8. 

That the G.P. is in the smallest terms possible is proved thus. 
- a, 6 are prime to one another, since the ratio a : i is in its lowest terms. 

[vii. 2»] 

Therefore o", #" are prime to one another ; so are «", ^ and, generally, 
*", ^. _ [vil. tj] 

Whence the g.p. is in the smallest possible terms, by viu. i. 

The Porism observes that, if there are h terms in the series, the 
extremes are («- i)th powers. 

',• .■• . 'iV) Proposition 3. ,^ ;\ 1^ '• , .v .'t 

// as many numbers as we please in continued proportion 
be the least of those which have the same ratio mith them, the 
extreines of them are prims to one anot/ter. 

Let as many numbers as we please, A, B, C, D, m con- 
tinued proportion be the least of those which have the same 
ratio with them ; 



■i-i < 

— E — F :> . • 1 

— O H K 

-L M N 



VIII. 3] PROPOSITIONS i, 3 349 

I say that the extremes of them A, D are prime to one 
aaother. 

For let two numbers E, F, the least that are in the ratio 
o^ A, B, C, D, be taken, [vii. 33] 

then three others G, H, K with the same property ; 

and others, more by one continually, [vm. 2] 

until the multitude taken becomes equal to the multitude of 
the numbers A, B, C, D. 

Let them be taken, and let them be L, M, N, 0. 

Now, since E, F are the least of those which have the 
same ratio with them, they are prime to one another, [vti. 22] 

And, since the numbers E, F by multiplying themselves 
respectively have made the numbers G, K, and by multiplying 
the numbers G, K respectively have made the numbers L, O, 

[vin. z, For.] 
therefore both G, /f and L, O are prime to one another, [vii. 27] 

And, since A, B, C, D are the least of those which have 
the same ratio with them, 

while Z,, Mi N, O are the least that are in the same ratio with 
A, B, C, D, 

and the multitude of the numbers A, B, C, D is equal to the 
multitude of the numbers L, M, N, O, 

therefore the numbers A, B, C D are equal to the numbers 
Li M, N, O respectively ; 

therefore A is equal to L, and D to O. 
And Z., O are prime to one another, 
• Therefore A, D are also prime to one another. 

Q, E, D, 

The proof consists in merely equating the given numbers to the terms of 
a series found in the manner of viii. 2. 

\i a, b,c, ... k {n terms) be a geometrical progression in the lowest terms 
having a given common ratio, the terms must respectively be of t>e form 

found by viii. 2, where a : j9 is the ratio a • 6 expressed in its lowest terms, so 

that a, J8 are prime to one another [vn. 21], and hence «""', fi*~^ are prime 

to one another [vii. 27], ., . 

But the two series must be the same, so that 

« = «■-', * = ;8"-' 



350 BOOK VIII [viit. 4 

Proposition 4. 

Given as many ratios as we please in leasi numbers, to find 
numbers in continued proportion which are ike least in the 
given ratios. 

Let the given ratios in least numbers be that of A to B, 
s that of C to D, and that q{ E to F\ 
thus it is required to find numbers in continued proportion 
which are the least that are in the ratio of A to B, in the 
ratio of C to D, and in the ratio of E to F, 



A— B ■•' 

D 

E F 

«— 5 

H 

M !i 

P L 



Let G, the least number measured by B, C, be taken. 

in And, as many times as B measures G, so many times also 
let A measure ^, 

and, as many times as C measures G, so many times also let 
D measure /C, 

Now E either measures or does not measure K. 
15 First, let it measure it. 

And, as many times as E measures A*, so many times let 
E measure L also. 

Now, since A measures // the same number of times that 
jff measures G, 
*> therefore, as y^ is to B, so is // to G. [vn. Def. io, vil 13] 

For the same reason also, 

as C is to D, so is G to K, 
and further, as ^ is to /^, so is A' to Z. ; 
therefore If, G, K, L are continuously proportional in the 
«s ratio of A to B, in the ratio of C to D, and in the ratio of E 
XoF. 

I say next that they are also the least that have this 
property. 



viii. 4] PROPOSITION 4 3S> 

For, if H, G, K, L are not the least numbers continuously 
30 proportional in the ratios of A to B^ of C to D, and of B 
to F, let them be N, 0, M, P. :- • j »., : 

Then since, as A is to B, so is N to O, 
while ^, ^ are least, 

and the least numbers measure those which have the same 
35 ratio the same number of times, the greater the greater and 
the less the less, that is, the antecedent the antecedent and the 
consequent the consequent ; 

therefore B measures 0. [vn. ao] 

' , .1 

For the same reason 
40 C also measures O; ;• 

therefore B, C measure O ; 

therefore the least number measured by B, C will also 
measure O. [vii. 35] 

But G is the least number measured by J9, C ; 
45 therefore G measures O, the greater the less : 
which is impossible. 

Therefore there will be no numbers less than H, G, K, L 
which are continuously in the ratio of A to B, of C to D, and 
oiBxaF. 
Sf> Next, let £ not measure J^. 





A 

B 


Q — 



E 

F 


.,, , 1 , 






0- 




H 




t 1 < 




K — 


















M 

































N — 
P — 























Let M, the least number measured by B, K, be taken. 

And, as many times as K measures M, so many times let 
H, G measure A', respectively, 

and, as many times as B measures M, so many times let F 
ss measure P also. 

Since H measures A'' the same number of times that G 
measures O, 
therefore, as /T is to 6^, so is A^ to O, [vn. 13 and Def, so] 



3S2 BOOK VIII [VIII. 4 

But, as ^ is to 6^, SO is /4 to ^ ; 
60 therefore also, as A is to B, so is A'^ to O. 1 io 1 . 1 ■• 

For the same reason also, 

as C is to D, so is O to M. 
Again, since ^ measures ^ the same number of times that 
F measures P, 
65 therefore, as ^ is to /^ so is ^ to /* ; [vu. 13 and Def, zo} 

therefore A'^, O, M, P are continuously proportional in the 
ratios of A to B, of C to /?, and of E to F, 

I say next that they are also the least that are in the ratios 
A:B, C:D. E:F. 
70 For, if not, there will be some numbers less than A^, O, 
M, P continuously proportional in the ratios A\B, C.D, 
E:F , , 

Let them be Q, R, S, T. 
Now since, as Q is to R, so is A to B, 
75 while A, B arc least, 
and the least numbers measure those which have the same 
ratio with them the same number of times, the antecedent the 
antecedent and the consequent the consequent, [vii. jo] 

therefore B measures R. 
80 For the same reason C also measures R ; 
therefore B, C measure R. 

Therefore the least number measured by B, C will also 
measure R. [vii. 35] 

But G is the least number measured by ^, C\ 
is therefore G measures R. 

And, as G is to R, so is A" to ,$" : [vii. 13J 

therefore K also measures S. 

But E also measures S; 
therefore E, K measure S, 
90 Therefore the least number measured by E, K will also 
measure S. [vn. 35] 

But M is the least number measured by E] K \ 
therefore M measures 5, the greater the less : 
which is impossible. 
9S Therefore there will not be any numbers less than A', O, 
Mt P continuously proportional in the ratios of A to B, of 
C to D, and of E to F; 



VIII. 4] PROPOSITION 4 353 

therefore N, O, M, P are the least numbers continuously 
proportional in the ratios A:B,C:D,E:F. q. e. d. 

So, 71, 09< the ratios A : B, C : D, B : F. TKu mbbreruted expression is in the 
Greek dI AB, TA, EZ \V- 

The terid " in continued proportion " is here not used in its proper sense, 
since a geometrical progression is not meant, but & series of terms each of 
which b^rs to the succeeding term a given, but not the same, ratio. 

The proposition furnishes a good example of the cumbrousness of the 
Greek method of dealing with non-determinate numbers. The proof in fact 
is not easy to follow without the help of modem symbotical notation. If 
this be used, the reasoning can be made clear enough. 

Euclid takes three given ratios and therefore requires to ^tsAfour numbers. 
We will leave out the simpler particular case which he puts first, that nameljr 
in which B accidentally measures K, the multiple of D found in the first few 
lines ; and we will reproduce the general case with Mr« ratios. 

Let the ratios in their lowest terms be 

; ^ a:i,e:d,e:/ .Tjiir 

Take li, the l.c.u. of i, e, and suppose that , ; \ ' , .' 

/, = mi — ne. 
Form the numbers ma, mli \, rut, 1 . . ,•.■ 

= nc\ ■ I ■ 

These are in the ratios of ji to d and of <r to rf respectively. ■.)•'• 'I 
Next, let /, be the l.c,m. of nd, e, and let ; ,'. J i 

4 =/«rf = qe. 
Now form the numbers 

pma, pmb \ , ptid \ , qf^ 
=pfie f =ge I 
and these are the four numbers required. 

If they are not the least in the given ratios, let 

f' y> ^' « , >\ ... ^^. V. .l.o'->i1| 
be less numbers in the given ratios. _. , 

Since ii : i is in its lowest terms, and '"''"''■ '-""' •' '"■ 

a \ b~x : y^ 

i measures V. ,h , . 

Similarly, since e : a=^ : z, 

t measures >■, 

Therefore /j, the L.C.M. of l>, f, measures >. 

But /, xnd\~e:d\^y: 

Therefore nd measures z. 

And, since e :f~z ; u, 

e measures z. 

Therefore 4i the L.C.M. of nd, e, measures z : which is impossible, since 
i</j or pnd. 

The step (line 86) inferring that G : H = K ; S 'n of course alternando 
from G:K[=C: D\ = Ji . S. 

It will be observed that viii. 4 corresponds to the portion of vi, 33 which 
shows how to compound two ratios between straight lines. 





>i. i-{'A 


-:> ti.H 


X. 




■ .-•ll.ii 


[ r^'i 


■ 'h 





354 BOOK VIII [viii. 5 



Proposition 5, 

Plane numbers have to one another the ratio compounded 
of the ratios of their sides. 

Let -4, ^ be plane numbers, and let the numbers C, D 

be the sides of A, and E, F oi B\ 

s I say that A has to B the ratio com- g 

pounded of the ratios of the sides. -. ^ 

For, the ratios being given which C _e — p 

has to E and D to F, let the least q 

numbers G, HyKih^t are continuously ^ 

10 in the ratios C\E,D:Fh^ taken, so ^ 

that, ^ 

as C is to E, so is G to Hy 
and, as D is to F, so is H to K. [viii. 4] 

And let D by multiplying E make L. 
IS Now, since D by multiplying C has made A, and by 
multiplying £" has made L, 
therefore, as C is to -£", so is ^ to Z,. ,. [vii. ij] 

But, as C is to ^, so is (7 to //^ ; 
therefore also, as G is to H, so is A to L. 
20 Again, since E by multiplying Z? has made L, and further 
by multiplying F has made ^, 
therefore, as /? is to /s so is Lkq B. - [vii. 17] 

But, as Z? is to F^ so\% H to K\ 
therefore also, as H is to K, so is L to ^. 
*s But it was also proved that, 

as fz is to H, so is ^ to Z. ; 
therefore, ex aeguali, 

as G^ is to K, so is A to B. [vn. 14] 

But G has to K the ratio compounded of the ratios of the 
30 sides ; 
therefore A also has to B the ratio compounded of the ratios 
of the sides. Q. e. d. 

I, 5, 19, 31. compounded of the ratios of Ibeir sides. As in v[. 13, the Greek 
has the less exact phitue, " cornpounded of their sides." 

If a = ed, b = tf, 

then a has to b the ratio compounded oi c le and d :/ 

Take three numbers the least which are continuously in the given ratios. 



viii. 5, 6] PROPOSITIONS 5, 6 JSS 

If / is the L.CII. of e, d and l=mt = nd, the three numbers are 

m/:, me \, nf. [vill. 4] 

~nd) 
Now dc:dt=c\c [vii. 17] 

= mc : me - me \ nd. 
Also td:tf=d\f [vii. 17] 

= ttd\nf. 
Therefore, ft* atguali, cd •.ef=mc : nf 

= (ratio compounded of 1: . e and d ;/). 
It will be seen that this proof follows exactly the method of vi. 23 for 
parallelograms. 



Proposition 6. 

// there be as many numbers as we please in continued 
proportion, and the first do not measure the second, neither 
will any other measure any other. 

Let there be as many numbers as we please. A, B, C, D, E, 
in continued proportion, and let A not measure B ; 
I say that neither will any other measure any other. 



-F 
— a 
H 



Now it is manifest that A, B, C, D, E do not measure 
one another in order ; for A does not even measure B. 

I say, then, that neither will any other measure any other. 

For, if possible, let A measure C. 

And, however many A, B, C are, let as many numbers 
F, G, H, the least of those which have the same ratio with 
A, B, C, be taken. [vn. 33] 

Now, since F, G, H are in the same ratio \-ith A, S, C, 
and the multitude of the numbers A, B, C is equal to the 
multitude of the numbers F, G, H, 
therefore, ex aequali, as A is to C, so is F to H. [vil 14] 



3jfir BOOK VIII . , [viri. 6, 7 

And since, as A is to ^, so is 7^ to G, a - ■ • ' ■ ■»> 
while A does not measure B, 

therefore neither does F measure G ; [vu. Def. ao] 

therefore F is not an unit, for the unit measures any number. 

"^om F, H are prime to one another. [vm. 3] 

And, as F is to H, so is ^ to C ; 
therefore neither does A measure C 

Similarly we can prove that neither will any other measure 
any other. 

Q. E. D. 

Let a^b,c...k\x,^ geometrical progression in which a does not measure h. 

Suppose, if possible, that a measures some term of the series, as / 

Take x,y, x, u, v, w the itcat numbers in the ratio a, b, c, d, e,f. 

Since x ■.y = a:b, 

and a does not measure b, ' 

X does not measure ^; therefore x cannot be unity. 

And, ex aequali, x : w = a :/. 

Now X, w are prime to one another. [viii. 3] 

Therefore a does not measure/ 

We can of course prove that an intermediate term, as b, does not measure 
a later term / by using the series b, e, d, e, f and remembering that, since 
b -.c-a; b, b does not measure c. 



Proposition 7. 

If there be as many numbers as we please in continued 

proportion, and the first measure the last, it tvUl measure the 
second also. 

Let there be as many numbers as we please, A, B, C, D, 
in continued proportion ; and 
let A measure D ; * 

I say that A also measures B, ^ 

For, if A does not measure ^' 

B, neither will any other of the ° — —" 

numbers measure any other. [vm. 6] 

But A measures D. 

Therefore A also measures B. ■," 



Q. £. D. 



An obvious proof by redudia ad absurdum from vm. 6, 



VHI. 8] PROPOSITIONS 6—8 ^^. 

I. 

Proposition 8. • ^' 

If between two numbers there fall numbers in continued 

proportion with them, then, however many numbers fall between 
them in continued proportion, so many will also fall in con- 
tinued proportion between the numbers which have the same 
ratio with the original numbers. 

Let the numbers C, D fall between the two numbers A, 
B in continued proportion with them, and let E be made in 
the same ratio to ^ as ^ is to ^ ; 

I say that, as many numbers as have fallen between A., B in 
continued proportion, so many will also fall between E, F in 
continued proportion. 



A e 

c M 

O N 

B F- ~- 

G 

""■ K 

L ' ■ - - " 

For, as many as A, B, C, D are in multitude, let so many 
numbers G, H, K, L, the least of those which have the same 
ratio with A, C, D, B, be taken ; [vn, 33] 

therefore the extremes of them (9j L are prime to one another. 

[vin. 3] 

Now, since A, C, D, B are in the same ratio with G, H, 
K,L, 

and the multitude of the numbers A, C, D, B is equal to the 
multitude of the numbers G, H, K, L, 
therefore, ex aequali, as A is to B, so is G to L. [vii. 14] 

But, as .(4 is to B, so is E to E; 
therefore also, as G is to Z, so is ^ to A ' 

But G, L are prime, "'~ 

primes are also least, [vn. si] 

and the least numbers measure those which have the same 
ratio the same number of times, the greater the greater and 
the less the less, that is, the antecedent the antecedent and the 
consequent the consequent. [vn. ^o] 



3Sg BOOK VIII [vui. 8, 9 

Therefore G measures S the same number of times as L 
measures F. 

Next, as many times as G measures E, so many times let 
/f, K also measure M, N respectively ; 
therefore <7, H, K, L measure E, M, N, F the same number 
of times. 

Therefore G, H, K, L are in the same ratio with E, M, 
N, F, [vn, Def. 20] 

But G, H, K, L are in the same ratio with A, C, D, B ; 
therefore A, C, D, B are also in the same ratio with E, M, 
N, F , 

But A, Ct D, B are in Continued proportion ; 
therefore E, M, N, FAre also in continued proportion. 

Therefore, as many numbers as have fallen between A, B 
in continued proportion with them, so many numbers have also 
fallen between E, F in continued proportion, 

Q, E. D. 
t. blL The Gi«k word is iiarhmir, " Ml in " = "can be interpolited." 

If a:6 = e:/, and between a, b there are any number of geometric 
means t, d, there will be as 'many such means between f, /. 

Let 0, fi, y, ■•■, S be the least possible terms in the same ratio as a, 
c, d,...b. 

Then o, 8 are prime to one another, [vtu. 3] 

and, ex atquali, a:% = a\b 

= €:f. 
Therefore t = ima,ys: m%, where m is some int^er. [vii. 10] 

t Take the numbers ma, mjS, my, ... mS. 

This is a series in the given ratio, and we have the same number of 
geometric means between ma, mS, or e,/, that there are between a, b. 

Proposition 9. 

If two numbers be prime to one another, and numbers fall 
between them in continued proportion, then, however many 
numbers fall between them in continued proportion, so many 
will also fall between each of them and an unit in continued 
proportion. 

Let A, B be two numbers prime to one another, and let 
C, D fall between them in continued proportion, 
and let the unit E be set out ; 
I say that, as many numbers as fall between A, B in con- 



VIII. 9] PROPOSITIONS 8, 9 399 

tinued proportion, so many wi]l also fall between either of 
the numbers A, B and the unit in continued proportion. 

For let two numbers F, Gy the least that are in the ratio 
of A, C, D, B, be taken, 

three numbers H, K, L with the same property, 

and others more by one continually, until their multitude is 
equal to the multitude of ^, C, D, B. [vni. a] 



A 




H 







ic • ■ •' 


D 




L • 


B 


E- 
F- 
Q— 










n -1 





. 


P 



Let them be taken, and let them be M, N, O, P. 

It is now manifest that F by multiplying itself has made 
H and by multiplying // has made M, while G by multiplying 
itself has made L and by multiplying L has made P. 

[viii. 3, Por.] 

And, since M, N, O, P are the least of those which have 
the same ratio with F^ G, 

and A, C, D, B are also the least of those which have the 
same ratio with F, G, [viii. i] 

while the multitude of the numbers M, N, C, P is equal to the 
multitude of the numbers A, C, D, B, 

therefore M, N, O, P are equal to A, C, £>, B respectively ; 

therefore Af is equal to A, and P to B. ,,. 

Now, since Fhy multiplying itself has made //, -■■'-■^ 

therefore F measures // according to the units in F. 

But the unit F also measures F according to the units in it; 

therefore the unit F measures the number F the same number 
of times as F measures If, 

Therefore, as the unit F is to the number F, so is F to ff. 

[vii. Def. »o] 

Again, since F by multiplying // has made M, 
therefore // measures M according to the units in F, v«.> 



36o BOOK VIII [viii. 9, lo 

But the unit £ also measures the number F according to 
the units in it ; 

therefore the unit B measures the number F the same number 
of times as // measures M. 

Therefore, as the unit £ is to the number F,so\s// to M. 

But it was also proved that, as the unit F is to the number 
F, so is /^ to /^ ; 

therefore also, as the unit F is to the number F, so is F to If, 
and // to M. 

But Jtf is equal \ja A \ 
therefore, as the unit E is to the number /", so is F to H, 
and H to A. 

For the same reason also, 
as the unit E is to the number G, so is C to Z and L to B. 

Therefore, as many numbers as have fallen between A, 
B in continued proportion, so many numbers also have fallen 
between each of the numbers A, B and the unit E in continued 
proportion, 

Q. E. D. 

Suppose there are n geometric means between a, b, Iwo numbers prime to 
one another ; there are the same number (n) of geometric means between i 
and a and between i and b. 

If c, d... are the n means between a, b, 

a, i, d ... b 
are the least numbers in that ratio, since a, b are prime to one another, [viii. ij 

The terms are therefore respectively identical with 

a"*', a-^, d'-'jS' ... Q^, iS"*', 

where o, ^ is the common ratio in its lowest terms. [vni. a. For.] 

Thus tf = <i"+', * = ^«+>. 

Now I Lffl = a I a' = a' : q'... =a» :a"*', 

and I:j8 = j3:^ = ^:j9'...=j8-;j8-*-; 

whence there are « geometric means between i, a, and between i, b. 



Proposition to. 

If numbers fall between each of two numbers and an unit 
in continued proportion, however many numbers fall between 
each of tkem and an unit in continued proportion, so many 
also will fall between the numbers themselves in continued 
proportion. 



VIII. lo] PROPOSITIONS 9, lo 361 

For let the numbers D, E and F, G respectively fall 
between the two numbers A, B and the unit C in continued 
proportion ; 

I say that, as many numbers as have fallen between each of 
the numbers A, B and the unit C in continued proportion, so 
many numbers will also fall between A, B in continued pro- 
portion. 

For let D by multiplying F make H, and let the numbers 
D, F by multiplying Ji make K, L respectively. 



.■^ i.^' 



c — 

D— 

E — 

F 

Q 



A- 








H 

K 

L 


I ' t 



Now, since, as the unit C is to the number D.sois D to £, 

therefore the unit C measures the number D the same number 
of times as D measures E. [vn. Def. 10] 

But the unit C measures the number £f according to the 
units in D ; 

therefore the number D also measures E according to the units 
inZ?; 

therefore I? by multiplying itself has made E. 

Again, since, as C is to the number D, so is E to A, 

therefore the unit C measures the number £> the same number 
of times as E measures A, 

But the unit C measures the number D according to the 
units in D ; 

therefore E also measures A according to the units in D ; 

therefore D by multiplying E has made A. - ' ' 

For the same reason also 

F by multiplying itself has made G, and by multiplying G has 
made B. 

And, since D by multiplying itself has made E and by 
multiplying F has made //, 

therefore, as Z> is to F, so is E to If. [vil 17] 



3«a BOOK Vm [vin. lo 

For the same reason also, -: .. ' 

as D is to /% so is // to C • ' " [vii. i8] 

Therefore also, as £ is to N, so h N to G. 

Again, since D by multiplying the numbers £, H has 
made A, K respectively, 
therefore, as E is to H, so is ^ to A'. ' [vii. 17] 

But, as E is to H, so\s D 10 E; 
therefore also, as D is to E, so is A to K. 

Again, since the numbers D, E by multiplying H have 
made K, L respectively, 
therefore, as D is to F, so is K to L, [vn. 18} 

But, as D is to F, ^q 'v& A to K \ 
therefore also, as ^4 is to K, so is K to L. 

Further, since F by multiplying the numbers //, G has 
made L, B respectively, 
therefore, as H is to G, so is L to B. [vil 17] 

But, as // is to G, so is D to A; 
therefore also, as /? is to 7% so is Z, to A • ' ■• ' • ' 

But it was also proved that, 

as D is to E, so is ^ to ^ and K to L\ 
therefore also, as A is to K, so is A" to Z and L to B. 

Therefore A, K, L, B are in continued proportion. 

Therefore, as many numbers as fall between each of the 
numbers A, B and the unit C In continued proportion, so 
many also will fall between A, B\u. continued proportion. 

Q. E, D, 

If there be n geometric nutans between i and a, and also between i and 
i, there will be « geometric means between a and b. 

The proposition is the converse of the preceding. 

The n means with the extremes form two geometric series of the fonn 
I, (I, a' ... a", 0"+', 
I, j8, ;8'...^, /8-+'. 
where a"^ ' = n, j3"+' = b. 

By multiplying the last term in the first line by the first in the second, 
the last but one in the first line by the second in the second, and so on, we 
get the series 

and we have the n means between a arid b. 

It will be observed that, when EucUd says " Fsr tke sami naion also, as 
i? is to /J so is .ff to G,'' the reference is really to vn. iS inst^id of vii. 17. 



viii. lo, ii] PROPOSITIONS lo, II 363 

He infers narady that Dx F: Fx F=D : F. But since, by vn, 16, the 

order of multiplication is indifferent, he is practically justified in saying " for 
the same reason." The same thing occurs in later propositions. 

Proposition ii. 

Between two square numbers there is one mean proportional 
number, and the square has to the square the ratio duplicate 
of that which the side has to the sieU. , , 

Let A, B\y& square numbers, " ' 

and let C be the side of W, and D o( B; 

I say that between A, B there is one mean proportional 

number, and A has to B the ratio 

duplicate of that which C has to D. a 

For let C by multiplying D make E. b ~ 

Now, since .<^ is a square and C is o □ 

its side, £ 

therefore C by multiplying itself has ,^ 

made A. . . . ' -i 

For the same reason also • • t' 

D by multiplying itself has made B. 

Since then C by multiplying the numbers C, D has made 
A, E respectively, 
therefore, as C is to /?, so is ^ to E, , [vii. 1 7] 

For the same reason also, 

as C is to /?, so is ^ to ^. • [vn, iS] 

Therefore also, as A is to E^ so is E to B. 

Therefore between A, B there is one mean proportional 
number. 

I say next that A also has to B the ratio duplicate of 
that which C has to D, 

For, since A, E, B are three numbers in proportion, 

therefore A has to B the ratio duplicate of that which A has 
to E. [v. Def. 9] 

But, as A is to E, so is C to D, 

Therefore A has to B the ratio duplicate of that which 
the side C has to D. q. e. d. 

According to Nicomachus the theorems in this proposition and the next, 
that two squares have one geometric mean, and two cubes two geometric 
means, betweerf them are Platonic. Cf. TimamSt 32 a sqq. and the note 
thereon, p. 294 above. 



'$64 BOOK VIII [viii. II, 13 

a*, ^ being two squares, it is only necessary to form the product ai and 
to prove that 

a", ai, ^ 
are in geometrica) progression. Euclid proves that 

^ '. ai = ai : ff' 
by means of vii. 17, 18, as usuaL 

In assuming that, since a* is to ^ in the duplicate ratio of a* to at, a* is 
to P in the duplicate ratio of a to 6, Euclid assumes that ratios which are 
the duplicates of equal ratios are equal. This, an obvious inference from 
V. 33, can be inferred just as easily for numbers from vii. 14. 

'.1 ;i>lr *n» "'J J Jirf iWli. 

Proposition 12. 

Between two cube numbers there are two mean proportional 
numbers, and tlie cube has to the cube the ratio triplicate of that 
which the side has to the side. 

Let A, B\>^ cube numbers, 
and let C be the side of A, and D oi B; 
I say that between A, B there are two mean proportional 
numbers, and A has to B the ratio triplicate of that which C 
has to /?. 



A- 






ttt K 


' '.''' 






C — 
















■A 




D 


K 




.» 





For let C by multiplying itself make £", and by multiplying 
D let it make F; 

let D by multiplying itself make G, 

and let the numbers C, D by multiplying F make H, K 
respectively. 

Now, since .^ is a cube, and C its side, - ... « •« li 

and C by multiplying itself has made E, 
therefore C by multiplying itself has made E and by multiply- 
ing E has made A, 

For the same reason also 
D by multiplying itself has made G and by multiplying G has 
made B. 

And, since C by multiplying the numbers C, D has made 
E, /^respectively, 
therefore, as C is to D, so is E to F, .... ■ [vu. ij] 



VIII. li, 13] PROPOSITIONS 11—13 3*5 

For the same reason also, 

as C is to /?, so is .^ to G. [vii. iSJ 

Again, since C by multiplying the numbers E, F has 
made A, H respectively, 
therefore, as E is to F, so is A to H, ' [vii. 17] 

But, as >£' is to F, so is C to /?. 

Therefore also, as C is to D, so is A to If. 

Again, since the numbere C, D by multiplying F have 
made H, K respectively, 
therefore, as C is to D, so is H to K, [vii. 18] 

Again, since D by multiplying each of the numbers F, G 
has made K, B respectively, 
therefore, as F is to G, so is K to B. [vii, 17] 

But, as /^ is to G, so is C to Z? ; 
therefore also, as C is to /?, so is Ava H,H to K, and K to B. 

Therefore H, K are two mean proportionals between A, B. 

1 say next that A also has to B the ratio triplicate of that 
which C has to D, 

For, since A, H, K, B are four numbers in proportion, 
therefore A has to B the ratio triplicate of that which A has 
to H, [v. Def. 10] 

But, as y^ is to Ht so is ^ to Z* ; 
therefore A also has to B the ratio triplicate of that which C 
has to D. 

Q. E. D, 

The cube numbere if, fi being given, Euclid forms the products n^b, a^ 

and then proves, as usual, by means of vii, 17, 18 that 

are in continued proportion. 

He assumes that, since a* has to ^ the ratio triplicate of o* : <^b, the 
ratio a' : ^ is triplicate of the ratio a : b which is equal to a* : (?b. lliis 
is again an obvious inference from vii. 14. 

Proposition 13. 

If there be as many numbers as we please in continued 
proportion, and each by multiplying itself make some number, 
the products ■mill be proportional ; and, if the original numbers 
by multiplying the products make certain numbers, the latter 
will also be proportional. 



0St Tt- BOOK VIII [viii. 13 

Let there be as many numbers as we please, A, B, C, in 
continued proportion, so that, as -(4 is to Ji, so is /? to C; 

let A, B, Chy multiplying themselves make D, E, F, and by 
multiplying D, E, /^let them make G, H, K\ 

I say that D, E, F and G, H, K are in continued proportion. 



A 


a 


B 


H 




K 






M 






_ 


N 








p 









Q 



>• For let A by multiplying B make L, • • .'i ''•• 

and let the numbers A, B by multiplying L make M. N 
respectively. 

And again let B by multiplying C make O, ' ''■ 

and let the numbers B, C by multiplying O make P, Q 
respectively. 

Then, in manner similar to the foregoing, we can prove 
that 

D, L, E and G, M, N, H are continuously proportional in the 
ratio of A to B, 

and further £, O, F and H, P, Q, K are continuously propor- 
tional in the ratio of B to C. 

Now, as A is to B^ so is ^ to C; 

therefore D, L, E are also in the same ratio with E, O, F, 

and further G, M, N, H in the same ratio with H, P, Q, K. 

And the multitude of D, L, E is equal to the multitude of 

E, O. F, and that of G, M, N, H to that of H, P,Q,K; 

therefore, ex acgualt, 

as D is to -£", so is £■ to F, '''' ' 
and, as tr is to H, so is H to K. [vn. 14] 



VIII. 13, 14] PROPOSITIONS 13, 14 3«J 

U a,i, c ...he a. series in geometrical progression, then 

J J 13 J f ^''^ *l^ '" geometncal progression. 

Heiberg brackets the words added to the enunciation which extend the 
theorem to any powers. The words are "and this always occurs with the 
extremes " (jtai a<i -rtpi tov( ixpov; toGto tnififlairu). They seem to be rightly 
suspected on the samt: grounds as the same words added to the enunciation 
of viL 27. There is no allusion to them iii the proof, much less any proof 
of the extension. 

Euclid forms, besides the squares and cul^es of the given numbers, the 
products ad, t^fi, atf, be, t^c, ic\ When he says that " we prove in manner 
similar to the foregoing," he indicates successive uses of vii. 17, iS as 
in VIII. 13. 

With our notation the prcx)f is as easy to sec for any powers as for squares 
and cubes. 

To prove that n", ^, <*,.. are in geometrical progression, " "'"' ' 

Form all the means between a', *", and set out the series 
a*, a'-'/', a'-'lr ... air-', ff. 
The common ratio of one term to the next is a ; /S. 

Next take the geometrical progression 

*". tr-^ i'-V ... //c^-\ i", v 

the common ratio of which i& i : c. 

Proceed thus for all pairs of consecutive terms, ' ' 

Now a : 6 = fi : c= ... 

Therefore any pair of succeeding terms in one scries are in the same ratio as 
any pair of succeeding terms in any other of the series. ,., 

And the number of terms in each is the same, namely (h + i). 

Therefore, m atgiialiy 

a-:*- = i» :(■ = <:" :rf" = ... 



Proposition 14. 

1/ a square measure a square, the side will also measure 
the side ; and, if the side measure the side, the square will also 
measure the square. 

Let A, Bh& square numbers, let C, D be their sides, and 
let A measure B; 
I say that C also measures D. A 

For let C by multiplying D make E ; fl 

therefore A, E, B are continuously pro- — ^ " 

portional in the ratio of C to D. [vm. nj e 

And, since A, E, B are continuously 
proportional, and A measures B, ' "■ 

therefore A also measures E. '• • '"- [vm, 7] 



3«8 BOOK VIII [viiL 14, IS 

And, as A is to E, so is C-io D\ ' "" ' 
therefore also C measures Z?, [vii. Def, ao] 

Again, let C measure D\ • " 

I say that A also measures B. 

For, with the same construction, we can in a similar 
manner prove that A^ E, B are continuously proportional in 
the ratio of C to Z?. 

And since, as C is to D, so is A to E^ 
and C measures D, 
therefore A also measures E. [vii. Def. ao] 

And A, E, B are continuously proportional ; 

therefore A also measures B. 

Therefore etc. 

Q. E. D. 

If «* measures ^, a measure l> ; and, if « measures b, a* measures P. 
(1) *i', ab, i° are in continued proportion in the ratio of a to b. 

(viii. 7] 



Therefore, since 


0' measures ^, 


..,-_. 


a' measures ab. 


But 


d^ : ab = a \ b. 


Therefore 


a measures b. 



(2) since a measures b, a* measures ab. 

And a', ab, P are continuously proportional. 
Thus ab measures ^, 

And a' measures ab. 

Therefore «' measures i". 

It will be seen that Euclid puts the last step shortly, saying that, since 
(j^ measures ab, and a*, ab, ^ are in continued proportion, a* measures #*. 
The same thing happens in viii. 15, where the series of terms is one more 
than here. 

Proposition 15. 

If a cube number measure a cube number, the side will also 
measure the side ; and, if the side measure the side, the cube 
will also measure the cube. 

For let the cube number A measure the cube B, 
and let C be the side of A and D oi B \ 
I say that C measures D. 



VIII. is] propositions 14, IS 3«^ 

For let C by multiplying itself make E, 
and let D by multiplying itself make G ; 
further, let Cby multiplying D make f, '"*» --tt .-•;, . >'> 
and let C, D by multiplying F make H, K respectively, 

A 



o— ' ■'■; K ■' 

E — ' 

°ir~ "---^ ■' 

Now it is manifest that E, F, G and A, H, K, B are 
continuously proportional in the ratio of Cto D. [viii. 11, u] 

And, since A, H, K, B are continuously proportional, 
and A measures B, 
therefore it also measures H. c- > , ■ [""■ '] 

And, as y4 is to /f, so is C to Z? ; 1 ,. .. 
therefore C also measures D. [vii, Dct ao] 

Next, let C measure D ; '" "" " •^''' "^ '' _ 

I say that A will also measure B. 

For, with the same construction, we can prove in a similar 
manner that A, //, K, B are continuously proportional in the 
ratio of C to D. 

And, since C measures D, 
and, as C is to D, so is A to H, 

therefore A also measures H^ [vii, Def. ao] 

so that A measures B also, 

Q. E. D. 

If 0* measures V, a measures b \ and via versa. The proor is, mutatii 
mutandis, the same as for squares. 

(i) (f,a'6,ai^,i^tK continuously proportional in the ratio of a to * ; 
and a* measures ^. 

Therefore «• measures a*i ; [viii. 7] 

and hence a measures i, "'■ -fi. ••>(_■ 1 

(i) Since a measures i, a* measures ci*j. 

And, 0*, a*3, aJ*, ^ being continuously proportional, each term measures the 
succeeding term ; 
therefore a* measures ^. 



.Ml 1« VI. 



Sf9 BOOK VIII [viii. i6, 17 

-TV 

^ Proposition 16. i^^^ 

If a square numder do not measure a square number, neither 
will the side measure the side ; and, if the side do not measure 
the side, neither will the square measure the square. 

Let A, B be square numbers, and let C, Z? be their sides ; 
and let A not measure B ; 

I say that neither does C measure D. * 

For, if C measures D, A will also ^ 

measure B. [vni. 14] c 

But A does not measure B ; d 

therefore neither will C measure D. 

Again, let C not measure Z? ; . . ' 

I say that neither will A measure B. 

For, if A measures B, C will also measure D. [vin. 14} 

But C does not measure D ; 
therefore neither will ^ measure B. 

Q. E. D. 

If a' does not measure ^, a mil not measure b; and, if a does not 
measure h t? will not measure ^. 

The proof is a mere rtduciio ad absurdum using vin. 14. 

Proposition 17. " 

If a cube number do not measure a cube numher, neither 
will the side measure the side ; and, if the side do not measure 
the side, neither will the cube measure the cube. 

For let the cube number A not measure the cube 
number B, 

and let C be the side of A, and D a^ 

of^; 

1 say that C will not measure D. 

For if C measures Z), A will 
also measure B. [viii. 15] 

But A does not measure B ; 
therefore neither does C measure D, 

Again, let C not measure D ; 
I say that neither will A measure B. 



^ — 
— 
D 



viiL 17, 18] PROPOSITIONS 16—18 371 

For, \{ A measures B, C will also measure D. [vm. 1 5] 
But C does not measure D ; 
therefore neither will A measure B. 

Q. E, D. 

If a* does not measure ^, a will not measure i ; and viee versa. 
Proved by reducHo ad absurdum employing viii. 15, 



Proposition 18. 

Between two similar plane numbers there is one mean 
proportional number ; and the plane number has to the plane 
number the ratio duplicate of that which the corresponding 
side has to the corresponding side. 

Let A, B be two similar plane numbei^, and let the numbers 
C, D be the sides of A, and E, F of B. 

K ' O 

B D 

E 

, . . F 



Now, since similar plane numbers are those which have 
their sides proportional, [vii. Def. «i] 

therefore, as C is to D, so \% E x.o F. ' ■ '' ' 

I say then that between A, B there is one mean propor- 
tional number, and A has to B the ratio duplicate of that 
which C has to E^ or D to F, that is, of that which the corre- 
sponding side has to the corresponding side. 

Now since, as C is to D, so is E to F, 
therefore, alternately, as C is to E, so is D to F. [vn, 13} 

And, since A is plane, and C, D are its sides, 
therefore D by multiplying C has made A. 

For the same reason also 
E by multiplying F has made B. 

Now let D by multiplying E make G. 

Then, since D by multiplying C has made A, and by 
multiplying E has made G, 
therefore, as C is to E, so is A to G. [vn. 17] 



37a " BOOK VIII [viii. i8 

But, as C is to £", SO is Z? to /^; • " ' 

therefore also, as /? is to J^, so is A to G. ''*' 

Again, since £ by multiplying D has made G, and by 
multiplying F has made B, 

therefore, as Z> is to /% so is C to B. '" \ " [vii. 17] 

But it was also proved that, 

as Z' is to F, so\% A to G; 
therefore also, as A is to (7, so is G to B. 

Therefore A, G, B are in continued proportion. 
'■•'• Therefore between A, B there is one mean proportional 
number. •^ 

I say next that A also has to B the ratio duplicate of 
that which the corresponding side has to the corresponding 
side, that is, of that which C has to £" or Z> to F. 

For, since A, G, B are in continued proportion, 

A has to B the ratio duplicate of that which it has to G. 

[v. Def. 9] 

And, as A is to G, so is C to E, and so is D to F. 
Therefore A also has to B the ratio duplicate of that which 
C has to £■ or Z? to F. 

Q. E. D, 

\i ab, <rf be " similar plane numbers," i.e. products of factors such that 
a • b = c ; d, 

thciie is one mean proportional between ai and at; and ab 'n to ed m the 

duplicate ratio of a to 1: or of ^ to ^. 

Fonn the product be (or ad, which is equal to it, by vii. 19). 

Then ' ab, be] , cd 



ab, be\, 
= ad)' 



is a series of terms in geometrical progression. 
For a : b=c •■ d. 

Therefore a:c = b:d. [v". 13] 

Therefore ab : be = be : cd. ' [vil. 17 and 16] 

Thus be (or ad) is a geometric mean between ah, cd. ' , 

And ab is to<rfin the duplicate ratio of ab to be or of be to <rf, that is, of 
d to f or of ^ to d. 



vm. 19] PROPOSITIONS 18, 19 jyj 

Proposition 10, 

Between two similar solid numbers there fall two mean 
proportional numbers; and the solid number has to the similar 
solid number the ratio triplicate of that which the corresponding 
side has to the corresponding side. 

Let j4, B he two similar solid numbers, and let C, D, E 
be the sides of A, and F, G, H of B. 

Now, since similar solid numbers are those which have 
their sides proportional, [vu. Def. si] 

therefore, as C is to /?, so is Fxa G, ••: I'l : '; 

and, as D is to ^, so Is 6^ to ff. 

I say that between A, B there fall two mean proportional 
numbers, and A has to B the ratio triplicate of that which C 

has to F, D to G, and also E to H. 

■ '■' \ . ■■•■ 



A- 

B 

C- F- N- 

0- o- 

E— H 

K— 

I ■ 

M 



For let C by multiplying D make K, and let F by 
multiplying G make X. 
■> Now, since C, D are in the same ratio with F, G, 

and /C is the product of C, V, and L the product of F, O, 
K, L are similar plane numbers ; [vn, Def. ai] 

therefore between K, L there is one mean proportional number. 

. , . [vin. 18] 

• Let it be ^ • ' '" - " ' ' 

Therefore M is the product of D, F, as was proved in the 

theorem preceding this. [vm. 18] 

Now, since D by multiplying C has made K, and by 

multiplying ^ has made M, 

therefore, as C is to /^ so \^ K Xa M. [vn. 17] 

But, as K is to M, so is M to L. 

Therefore K, M, L are continuously proportional in the 
ratio of C to F, 



f^4 . ot .t BOOK VIII [viii. 19 

And since, as C is to D, so is F to G, 
alternately therefore, as C is to /% so is Z? to G. [vii. 13] 

For the same reason also, 

as D is to G, so is E to H. 

Therefore K, M, L are continuously proportional in the 
ratio of C to F^ in the ratio of D to G, and also in the ratio 
of E to H. 

Next, let E, H by multiplying M make A^, O respectively. 

Now, since ^ is a solid number, and C, D, E are its sides, 
therefore E by multiplying the product of C, D has made A. 

But the product of C, D is K; • . ■ - lis.- ; 

therefore E by multiplying K has made A. • ' • m= 

For the same reason also ' ' 

H by multiplying L has made B. 

Now, since E by multiplying A' has made A, and further 
also by multiplying M has made N^ 

therefore, as K is to M, so is j4 to N. [vii. 17] 

But, as K is to ^, so is C to F, D to £7, and also E to //■; 

therefore also, as C is to Z', Z? to G, and EtoH,ia\%A to A'^, 

Again, since £", ^ by multiplying M have made A^, O 
respectively, 

therefore, as ^ is to //, so is N to O. [vii. 18] 

But, as ^ is to H, so is C to /^ and D Xa G\ 
therefore also, as C is to /% Z? to G, and E to H,^a\% A to 
iV and A^ to a 

Again, since //^ by multiplying ^has made f?, and further 
also by multiplying L has made B, 

therefore, as ^ is to Z,, so is f? to B, [vil 17] 

But, as ^ is to Z, so is C to F, D to G, and ^ to H. 
Therefore also, as C is to F, D to G, and E io H, so not 

only is t? to ^, but also A to N and A'' to O. 

Therefore A, N,0, B are continuously proportional in the 

aforesaid ratios of the sides. 

I say that A also has to B the ratio triplicate of that which 
the corresponding side has to the corresponding side, that is, 
of the ratio which the number C has to F, or D to G, and 
also E to H. 



VIII. 19, ao] PROPOSITIONS 19, 20 375 

For, since A, N, O, B are four numbers in continued 
proportion, 

therefore A has to B the ratio triplicate of that which A has 
to N, [v. Def. ro] 

But, as y^ is to A'', so it was proved that C is to F, D to (?, 
and also E to H. 

Therefore ^ also has to ^ the ratio triplicate of that which 
the corresponding side has to the corresponding side, that is, 
of the ratio which the number C has to F, D to G^ and also 
E to H, Q. E. D. 

In other words, M a:b : c=d : e :/, then there are two geometric means 
between abc, def; and abc is to def in the triphcate ratio of a to d, or b to e, 
or c \,of 

Euclid first takes the plane numbers ab, dt (leaving out e, f) and foroos 
the product bd. Thus, as in viu. 18, 

ab, bd\ , de 
-eaj 
are three terms in geometrical progression in the ratio of a to d, or of i to e. 

He next forms the products of f,/ respectively into the mean bd. 

Then afic, ebd, fid, def 

are in geometrical progression in the ratio of a to ^ etc. 



Fm abc 

bd 
fbd 



: cbd^ab ; bd=a : d' 

•.fbd = c:f -. [vii. 17] 

•.def=bd:de=b:e ] 
And a : d-b : t = c -.f. 

The ratio of abc to def is the ratio triplicate of tliat of abc to ebd, ie. of 
that of a ta d etc. 

Proposition 20. 

If <me mean proportional number fall between two numben, 
the numbers will be similar plane numbers. 

For let one mean proportional number C fall between the 
two numbers A, B\ 
5 1 say that A, B are similar plane numbers. 

Let D, E, the least numbers of those which have the same 
ratio with A, C, be taken ; [vii. 33] 

therefore D measures A the same number of times that B 
measures C. [vu. no] 

10 Now, as many times as D measures A, so many units let 
there be in ^; r 

therefore F by multiplying Z? has made A, 
so that A is plane, and D, F are its sides. 



3.f6 BOOK VIII 'Vi [viii. ao 

Again, since />, £ are the least of the numbers which have 
IS the same ratio with C, B, 
therefore D measures C the same number of times that E 
measures B. [vn. 20] 

A' O 



B- 



F- 
Q- 



As many times, then, as E measures B^ so many units let 
there be in G\ 
ao therefore E measures B according to the units in G^; 
therefore G by multiplying E has made B. ' " 

Therefore B is plane, and E, G are its sides, 

Therefore A, B are plane numbers. 

I say next that they are also similar. - » 

«S For, t since F by multiplying D has made At and by 
multiplying E has made C, 
therefore, as D is to E, so is A to C, that is, C to B. [vii. 17] 

Again,t since E by multiplying F, G has made C, B 
respectively, 

30 therefore, as F is to G, so is C to B. [vn. 1 7] 

But, as C is to .ff, so is Z? to ^ ; 
therefore also, as D is to E, so is F to G. 

And alternately, as D is to F, so is E to G. [vn. 13] 

Therefore A, B are similar plane numbers; for their sides 

3S are proportional. q. e. d. 

J J. For, since F 17. C to B. The tent has clearly suffered oomiplion here. It 

is not neccssaty to inftr from otfaer facts th^t, ^ 2^ is to £, so \& A Ui C\ for this is part of 
the hypotheses (II. 6, 7). Again, there is no enplanation of the statement {1. 55) that ^ by 
multiplying E has made C. It is the statement and explanation of this latter fact which are 
atene wanted ; after which the proof proceeds as in 1. 18. We might therefore luiHtitute for 
It. J5 — »8 Ibe following, 

"For, since £ measures C the same number of times that D measures A [1. 8], that is, 
according to the units in F [1. loj, therefore F by multiplying E has made C. 

And, since E by moltiplying F, G," etc. etc. 

This proposition is the converse of vi 11, 18. \i a,e, b are in geometrical 
pn^ession, a, b are " similar plane numbers." 

Let a : j3 be the ratio a i ( (and therefore also the ratio f : i) in its lowest 
tenns. 

Then [vn. 20] 

a ~ ma, c - mfi, where m is some integer, ' ' ' ' 

f = wo, i = »ft where « is some integer. 



VIII. 30, *i] PROPOSITIONS JO, a I m 

Thus a, ^ are both products of two factors, i.e. plane. ■ , , , 
,. Again, a : fi = a : e = t : A ^ 

= m:n. ■<.,.. ,, [v". i8] 

Therefore, alternately, a:m='^:n, • [vii, 13] 

and hence ma, «j3 are similar plane numbers. 

[Our notation makes the second part still more obvious, for ^ =»(jS=«(x.] 

I Proposition 21. 

If two mean proportional numbers fali between ia)9 numbers, 
the numbers are similar solid numbers. 

For let two mean proportional numbers C, D fall between 
the two numbers A, B ; 
I say that A, B are similar solid numbers. 



Ill' 



A E- ^ ■ ■ •■• • 

B F •':■ ■' 

.. ^ c a .,/,'• 

D M- 

,\i I, . , «— . K- 

O L- 

M 

For let three numbers E, F, G, the least of those which 
have the same ratio with A, C, D, be taken ; [vii. 33 or vin. a] 

therefore the extremes of them £, G are prime to one another. 

[vni. 3] 

Now, since one mean proportional number F has fallen 
between £, G, 
therefore E, G are similar plane numbers. [vm. aoj 

Let, then, H, K be the sides of E, and L, M of G, 

Therefore it is manifest from the theorem before this that 
£■, F, G are continuously proportional in the ratio oi H to L 
and that of K to M. 

Now, since £, F, G are the least of the numbers which 
have the same ratio with A, C, D, 

and the multitude of the numbers E, F, G Is equal to the 
mtiltitude of the numbers A, C, D, 
therefore, ex aequali, as E is to G, so is A to /?. [vu. 14] 

But B, G are prime, 
primes are also least, [vii. ai] 

and the least measure those which have the same ratio with 



jifS BOOK Viri ' "^ fviii. 31 

them the same number of times, the greater the greater and 
the less the less, that is, the antecedent the antecedent and the 
consequent the consequent ; [vn. 20] 

therefore £ measures A the same number of times that G 
measures £>. 

Now, as many times as E measures A, so many units let 
there be in TV. 

Therefore JV by multiplying £ has made A. 

But £ is the product of //, K ; 
therefore N by muhiplying the product of J/, K has made A, 
,. Therefore A is solid, and H, K, N axe. its sides. ,^ . 

Again, since E, F, G are the least of the numbers which 
have the same ratio as C, D, B, 

therefore E measures C the same number of times that G 
measures B. 

Now, as many times as E measures C so many units let 
there be in O. 

Therefore G measures B according to the units in 0\ 
therefore O by muhiplying G has made B. 

But G is the product of L, M; 
therefore O by multiplying the product of L, M has made B, 

Therefore B is solid, and L. M, O are its sides ; 
therefore A, B are solid. 

I say that they are also similar. 

For since N, O by multiplying E have made A, C, 
therefore, as N is to O, so is A to C, that is, E to F. [vn. 18] 

But, as E is to F, so is H 10 L and K to M\ 
therefore also, as // is to Z, so is A" to ^ and N to O. 

And H, K, N are the sides of A, and O, L, M the sides 
of^. 

Therefore A, B are similar solid numbers. q, e. d. 

The converse of viu. 19. If a, e, d, b are in geometrical progression, a, b 
are "similar solid numbers." 

Let a, ;3, y be the least nunibers in the ratio of a, ^, d (and therefor* also 
of i-, rf, b). [vii. 33 or vui. »■ 

Therefore a, y are prime to one another, , .,.,., [vhi. 3^ 

They are also " similar plane numbers." ' [vni. 30 

Let a. = mn, y =pf, 

where m:n-p:q. ■.-■■-. 



viii. ii-ij] PROPOSITIONS zi—i3 j79 

Then, by the proof of viii, 20, 

a :j3 = m:/ = « : ^. 

Now, ex aegua/i, a ; J-a : y, [vil. 1 4} 

and, since a, y are prime to one another, 

B = m, d-ry, where r is an integer. 

But a = mn\ 

therefore a = rw«, and therefore a is " solid." 

Again, ex aequaii, c : &- a. : y, 

and therefore c = ja, b = sy, where J is an integer. 

Thus b = spg, and d is therefore " solid." 

Now a ; j3 = a : e = ra: so. 

= r:s. [vii. 18] 

And, from above, a: fi-m : p- n : g. 

Therefore r \ s = mip = » -.g, 

and hence a, b are similar sohd numbers. 

Proposition 22. 

/^ ^A^^e numbers be in continued proportion, and the first 
be sqxtare, the third will also be square. 

Let A, Bt C" be three numbers in continued proportion, 
and let A the first be square ; 
I say that C the third is also square. "' 

For, since between A, C there is one 

mean proportional number, B, 

therefore A, C are similar plane numbers. [viii. 30] 

But A is square ; " ' ' 

therefore C is also square. Q. e. d. 

A mere application of viu. zo to the particular case where one of the 
" similar plane numbers " is square. 

Proposition 23. 

I/four numbers be in continued proportion, and the first be 
cube, the fourth will also be cube. 

Let A, B, C, i? be four numbers in continued proportion, 
and let A be cube ; 



I say that D is also cube. * 

For, since between A, D there c 

are two mean proporti onalnumbers q 

B, C, 

therefore A, D are similar solid numbers, [vui. ai] 



|9q n- BOOK VIII >i'l [vui. 33—35 

But^ is cube; 1 .,k ..• r <■ 

therefore Z? is also cube. ' • 

Q, E, D. 

A mere application of viii. a i to the case where one of the " similar solid 
numbers " is a cube. _, , j 

., ,. t Proposition 34. ,. „jj 

// two numbers have to one another the ratio which a square 
number has to a square number, and the first be square, the 
second wiU also be square. 

For let the two numbers A, B have to one another the 
ratio which the square number C has 

to the square number D, and let j4 be A 

square ; B 

I say that ^ is also square. q 



For, since C, D are square, 
C, D are similar plane numbers. 

Therefore one mean proportional number falls between 
C, D. [viii. 18] 

And, as C is to Z?, so is .(4 to 5 ; 
therefore one mean proportional number falls between A, B 
also. [viii. 8] 

And j4 is square ; .■i'- 

therefore B is also square. ; , . [vm, 21] 

Q. E. D. 

If « ; # = ^' ; (f", and « is a square, then b is also a square. 

For f', dy have one mean proportionai (d. [vm. 18] 

TheKfoie a, b, which are in the same ratio, have one mean proportional. 

[vm. 8] 
And, since a is square, b must also be a square. , . [vm, 33] 

Proposition 25. 

If two numbers have to one another the ratio which a cube 
number has to a cube number, and the first be cube, the second 
will also be cube. 

For let the two numbers A, B have to one another the 
ratio which the cube number C has to the cube number D, 
and let A be cube ; 
1 say that B is also cube. ' ^ • • 1 >ii 



«ii. »5, a6] PROPOSITIONS 13— »fi 3«i 

For, since C, D are cube, 

C, /? are similar solid numbers. 

Therefore two mean proportional numbers fall between 
D. [viii. ig] 



C, D 



A E- 

B F- 

O ■ 



And, as many numbers as fall between C, D in continued 
proportion, so many will also fall between those which have 
the same ratio with them ; [vm. 8] 

so that two mean proportional numbers fall between A, B 
also. 

Let £, Fso fall. 

Since, then, the four numbers A, E, F, B are in continued 
proportion, 
and A is cube, 
therefore B is also cube. [vm. aj] 

Q. E. D, 

ir d : b = ('id', and a is a cube, then b is also a cube. 
For c', d* have two mean proportionals, [viil. 19] 

Therefore a, b also have two mean proportionals, [vin. 8] 

And a is a cube : 
therefore £ is a cube. [vm. 33] 



Proposition 26. 

Similar plane numbers have to one another the ratio which 
a square numier has to a square number. , .. 

Let A, B be similar plane numbers ; 
I say that A has to B the ratio which a square number has 
to a square number. 



l» .1 « 

c- 
D e- 



For, since A, B are similar plane numbers, 

therefore one mean proportional number falls between A, B, 

[vm. 18] 



^ '^- BOOK VIII [viii. a6, 37 

Let it so fall, and let it be C; —- ^- ' - 

and let /}, £, F, the least numbers of those which have the 
same ratio with A, C, B, be taken ; [vu, 33 or vm. 2] 

therefore the extremes of them D, F are square, fvm. i. Pot.] 

And since, as D is to F, so is A to B, 

and D, F are square, 

therefore A has to B the ratio which a square number has to 
a square number. 

Q. E. D. 

If a, ^ are similar "plane numbers," let ^ be the mean proportional 
between them, [vui. 18 

Take a, P,y the smallest numbers in the ratio of a, c, t. [vti. 33 or viii. 2 
Then a, y are squares. [vii[, j, Por.^ 

Therefore a, i are in the ratio of a square to a square. 



PrOI'OSITION 27. 

Similar solid numbers have to one another the ratio which 

a cube number has to a cube number. 

Let A, Bh^ similar solid numbers ; . . 

I say that A has to B the ratio which a cube number has to 
a cube number. 



A c- 

B D- 

E — F Q H 



For, since A, B are similar solid numbers, 

therefore two mean proportional numbers fall between A, B. 

[viii. 19] 
Let C, D so fall, 

and let E, F, G, H, the least numbers of those which have 
the same ratio with A, C, D, B, and equal with them in 
multitude, be taken ; [vii. 33 or vm, j] 

therefore the extremes of them E, H are cube. [vm. », Por.] 

And, as E\iXo H,^o\% A Xa B\ 

therefore A also has to B the ratio which a cube number has 

to a cube number. ,^ 

Q. E. D. 



vui. 37] PROPOSITIONS 26, 27 383 

The sime thing as viii. 26 with cubes. It is proved in the same way 
except that viii. 19 is used instead of viii. 18, 

The last note of an-Nairlzi in which the name of Heron is mentioned is 
on this proposition. Heron is there stated (p. 194 — 5, ed. Curtze) to have 
added the two propositions that, 

I. If two numbtri havt to ons another the ratio 0/ a square to a square, the 
numbers are similar f lane numbers ; 

1, If two numbers have to one another the ratio of a cube to a cube, the numbers 
are similar solid numbers. 

The propositions are of course the converses of viii. a 6, 27 respectively. 
They are easily proved 

(i) If a:b = c':<P, 

then, since theie is one mean proportional {ed) between c\ d*, 

[viii. 1 1 or 18] 
there is also orx mean proportional between a, b. [viii. 8] 

Therefore a, b are simitar plane numbers. [viti. 20] 

(1) is similarly proved by the use of viii. 12 or it^, viii. 8, vtit. 31. 

The insertion by Heron of the first of the two propositions, the converse 
of vm. a6, is fwrhaps an argument in favour of the correctness of the text of 
IX. 10, though (as remarked in the -note on that proposition) it does not give 
the easiest proof Cf Heron's extension ni vii. 3 tacitly assumed by Euclid 
in vii, 33. 

'.' lifi ". .'■\ Tfpf.-i ?Rfi ';•• .. -. ..; 'j.'i ' 

I, ..'. , 1 ' 'i .'I I,' V'-. ,.i I , »; ■ i •' .; I*}' 

* . ,1 •.<K..vJ •.! ' iTi-iL;ii.t(i la«" .r-'" -1 I'/'j.-' ' 






.{ <l-;-iW'-, ' ■ V. ,.'•■•('.1 I; JLl'i 






-■rajrt a 



M- 






BOOK IX. 



A- 
B- 
C- 



Proposition r. 

// two similar plane numbers by muUiplying one another 
make some number, the product will be square. 

Let A, B be two similar plane numbers, and let A by 
multiplying B make C; 
I say that C is square. 

For let A by multiplying itself 
make D. ^ 

Therefore /? is square. 

Since then A by multiplying itself has made D, and by 
multiplying B has made C, 

therefore, as ^ is to ^, so is Z? to C [vti. 17] 

And, since A, B are similar plane numbers, 

therefore one mean proportional number falls between A, B. 

[vm. 18] 

But, if numbers fall between two numbers in continued 

proportion, as many as fall between thetn, so many also fall 

between those which have the same ratio ; [vni. 8] 

so that one mean proportional number falls between D, Calso. 

And D is square ; 
therefore C is also square. [vm. la] 

Q. E. D. 

The product of two similar pkne numbers b a square. 
Let a, ^ be two similar plane numbers. 

Now « : i = fl" ; oi. [vii. 1 7" 

And between a, b there b one mean profwrtional. [vm, \i 

Therefore between «' : «# there is one mean proportional. [vm, 8' 

And <^ is square ; 
therefore ab is square. [vm. ai] 



IX. *, 3l PROPOSITIONS 1—3 ^ 

■ ■ . . : - ,"1 

Proposition 2, - . .. 1 

If two numbers by multiplying one another make a square 
number, they are similar plane numbers. 

Let A, B be two numbers, and let A by multiplying B 

make the square number C\ , . 

f I ' **!' ('■ est* ' 

I say that A, B are similar plane '^' ' 
numbers. b 

For let A by multiplying itself ^ 
make D ; ° 

therefore D is square. '^ " ' ' "' ■'"''>• 

Now, since ^ by multiplying itself has made D, and by 
multiplying B has made C, 
therefore, as ^ is to ^, so is Z) to C ' ^ '' '"■'-• [vn. 17] 

And, since D is square, and C is so also, 
therefore D, C are similar plane numbers. 

Therefore one mean proportional number falls between 
A C. [vin. 18] 

And, as D is to C so is .^ to ^ ; 
therefore one mean proportional number falls between v4,./? 
also. [viii. 8] 

But, if one mean proportional number fall between two 
numbers, they are similar plane numbers ; [vim. 20] 

therefore A, B are similar plane numbers. 

Q. E. D, 

ir oj is a square number, «, b are similar plane numbers, (The converse 
of IX. i<) 

For a ; b - a* : al^. [vn. ij] 

And a', ab being square numbers, and therefore similar plane numbers, 

they have one mean proportional. [vin. 18] 

Therefore a, b also have one mean proportional. [vni. 8] 

whence a. b are similar plane numbers. rvni. aol 

"" Proposition 3. 

If a cube number by multiplying itself make some number, 
the product vnll be cube. 

For let the cube number A by multiplying itself make B ; 
I say that B is cube. 



386 BOOK IX [ix. 3 

For let C, the side of ^, be taken, and let C by multiplying 
itself make D. 

It is then manifest that Cby muhiplying a 

D has made A. ^ 

Now, since C by multiplying itself has c- d — 

made D, 
therefore C measures D according to the units in itself. 

But further the unit also measures C according to the units 
in it; 
therefore, as the unit is to C, so is C to D. [vn, Def. ao] 

Again, since C by multiplying D has made A^ 
therefore D measures A according to the units in C. 

But the unit also measures C according to the units in it ; 
therefore, as the unit is to C so is Z? to -(4. 

But, as the unit is to C, so is C to Z? ; 
therefore also, as the unit is to C, so is C to D, and D to A. 

Therefore between the unit and the number A two mean 
proportional numbers C, D have fallen in continued proportion. 

Again, since A by multiplying itself has made B, 
therefore A measures B according to the units in itself. 

But the unit also measures A according to the units in it; 
therefore, as the unit \% \.<a A , ?^ '\% A to B. [vu. Def. 20] 

But between the unit and A two mean proportional numbers 
have fallen ; 

therefore two mean proportional numbers will also fall between 
A, B. {vtu. 8] 

But, if two mean proportional numbers fall between two 
numbers, and the first be cube, the second will also be cube. 

[vni. 23] 

And A is cube ; 
therefore B is also cube. ' Q. e. d. 

The product of t^ into itself, or a" , a', is a cube. 

For I : rt = a : fl' = a* : b'. 

Therefore between i and a' there aie two mean proportionals. 

Also I : a^ = u' : a* . o*. 

Therefore two mean proportionals fall between a' and «' . a*. [vui, 8] 

(U is true that viii. 8 is only enunciated of two pairs of numbers, but the 
proof is equally valid if one number of one pair is unity.) 

And a* is a cube number: 
therefore o^ . <t' is also cube. [viii. 13] 



IX. 4, 5] HROPOSITIONS 3—5 38J 

Proposition 4. 

If a cuie number by multiplying a cube number make some 
number, the product will be cube. 

For let the cube number A by multiplying the cube number 
B make C ; 
I say that C is cube. A 

For let A by multiplying^ e— — 

itself make D ; c 

therefore D is cube. [ix. 3] ° 

And, since A by muUtply- 
ing itself has made D, and by multiplying B has made C 
therefore, as A is to B, so is D to C [vii. ij] 

And, since A, B are cube numbers, 
A, B are similar solid numbers. 

Therefore two mean proportional numbers fall between 
A, B ; [viii. 19] 

so that two mean proportional numbers will fall between D, 
C also. [viii. 8] 

And D is cjbe; 
therefore C is also cube [vm. 13] 

Q. E. D. 

The product of two cubes, say «* . ^, is a cabe. 

For a' : *' = a'.o' : d*.*"." [vii. 17] 

And two mean proportionals fall between a*, f which are similar solid 

numbers, [vili. 19' 

Therefore two mean proportionals fall between 4' .i^, ^.y [viii. 8' 

B jt a* , o* is a cube : [ix. 3 

therefore a*. ^ is a cube. ' ' [vili, 23 

Proposition 5. 

// a cube number by multiplying^ any number make a cube 
number, Ike multiplied number will also be cube. 

For let the cube number A by multiplying any number B 
make the cube number C; 
I say that B is cube. /^ 

For let A by multiplying s « .<•' 

itself make D ; c 

therefore D is cube. [ix. 3] P^ 



^ BOOK IX [IX, 5, 6 

Now, since ^ by multiplying itself has made J?, and by- 
multiplying £ has made C, 
therefore, as >4 is to B, so is Z? to C. v^<fv,a ■:■'.*' [vn. 17] 

And since Z>, C are cube, ' •"'**'i»^, ►» 

they are similar solid numbers. 

Therefore two mean proportional numbers fall between 
D, C [vill, ig] 

And, as Z* is to C, so is ^ to ,5 ; 
therefore two mean proportional numbers fall between A, B 
also, [vin, 8] 

And A is cube ; 
therefore B is also cube, [viii. aj] 

If the product cfb is a cube number, b is cube. " ' 

By IX. 3, the product a',<i' is a cube. . V. ■ ,. • 

And tf , <^ : tfb = tf i b. [vn. 17] 

The first two terms are cubes, and therefore "similar sohds"; therefore 

there are two mean proportionals between them. [^i"- 19] 

Therefore there are two mean proportionals between a*, b. [vn:. 8J 

And 0* is a cube : 

therefore i is a cube number. [viiL 93] 

Proposition 6, 

If a nttmber by muUiplytng itself make a cube number, it 
will itself also be cube. 

For let the number A by multiplying itself make the cube 
number B ; 

I say that A is also cube, ■ '■""•'"""•I-". a 

. For let A by multiplying B make C. ^ 

Since, then, A by multiplying itself o 

has made B, and by multiplying B has 

made C, 

therefore C is cube. -.^- ■.u.av. -. -/w .. .< 

And, since A by multiplying itself has made B, 
therefore A measures B according to the units in itself. 

But the unit also measures A according to the units in it 
Therefore, as the unit is to A, so is A to B, [vn. Def. ao] 
And, since A by multiplying B has made C, 
therefore B measures C according to the units in A. 

But the unit also measures A according to the units in it. 



IX. 6, 7] 



PROPOSITIONS s— 7 



Therefore, as the unit is to A, so is B to C. 

But, as the unit is to A, sols A to B ; 
therefore also, as ^ is to B, so is ^ to C. . 

And, since B, C are cube, 
they are similar solid numbers. 

Therefore there are two mean 
between B, C. 

And, as B is to C, so is ^^ to B. 

Therefore there are two mean 
between A, B also. 

And B is cube ; 
therefore A is also cube. 



3«9 
[vii. Def. so] 



proportional 



proportional 



numbers 
[viii. 19] 

numbers 

[vm. 8] 



[cf. vm. 23] 

Q. E. D, 



If a" IS a cube number, a is also a cube. 

For I ; a = fl : a' = a' : a*. 

Now a\ (^ are both cubes, and therefore "similar solids "; therefore ther« 
:\je two mean proportionals between them. [vm. 19] 

Therefore there are two mean proportionals between a, a', [vm. 8] 

And a* is a cube : 
therefore a is also a cube number. [vm. 23] 

It will be noticed that the last step is not an exact quotation of the result 
of vtn. «3, because it is there the first of four terms which is known to be a 
cube, and the last which is proved to be a cube ; here the case is reversed. 
But there is no difficulty. Without inverting the proportions, we have only 
to refer to vm. 2r which proves that a, cf, having two mean proportionals 
between them, are two similar solid numbers ; whence, since a" is a cube, 
a is also a. cube. 



Proposition 7. 

Tf a composite number by multiplying any number make 
some number, the product will be solid. 

For let the composite number A by multiplying any number 

B make C\ 

f^, 

I say that C is solid. ^ 

For, since^ is composite, q 

it will be measured by some ^ ^ 

number. [vn. Def. 13] 

Let it be measured hy D\ 
and, as many times as D measures A, so many units let there 
be in B. 



|9l9 BOOK IX [ix. 7, S 

[o Since then D measures A according to the units in £", 
therefore E by multiplying D has made A. \ym Def. 15] 

And, since A by multiplying B has made C, 
and A is the product of D, E, 
therefore the product of D, E by multiplying B has made C. 

Therefore C is solid, and D, E, B are its sides. 

Q. E. D. 'X^ 

Since a composite number is the product of two factors, the result of 
multiplying tt by another number is to produce a ~n umber which is the 
product of three factors, i.e. a "solid number." 

; . ..I . . -. , . -.^ ,...-:), 

, , , Proposition ST. 

If as many numbers as we please beginning from an unit, be 
in continued proportion, the third from the unit will be square, 
as will also those which successively leave out one ; the fourth 
will be cube, as will also all those which leave out two; and the 
seventh will be at once cube and square, as will also those which 
leave out five. 

Let there be as many numbers as we please, A, B, C, D, 
B, F, beginning from an unit and in con- 
tinued proportion ; A 

I say that B, the third from the unit, is ^ 



square, as are also all those which leave p 

out one ; C, the fourth, is cube, as are g 

also all those which leave out two ; and p 

E, the seventh, is at once cube and 

square, as are also all those which leave out five. 

For since, as the unit \s to A , so is A to B, 
therefore the unit measures the number A the same number 
of times that A measures. .5, [vu. Def 26] 

But the unit measures the number A according to the 
units in it ; 
therefore A also measures B according to the units in A, 

Therefore A by multiplying itself has made B; 
therefore B is square. 1 . .• 

And, since B, C, Z? are in continued proportion, and B is 
square, 
therefore D is also square. [vtit. a 3] 



IX. 8] PROPOSITIONS 7, 8 39* 

For the same reason v. , t .1. .u m.- 

r IS also square. 

Similarly we can prove that all those which leave out one 

are square. 

I say next that C, the fourth from the unit, is cube, as are 
also all those which leave out two. 

For since, as the unit is to W, so is 3 to C, 

therefore the unit measures the number A the same number 
of times that B measures C. 

But the unit measures the number A according to the units 
in A ; 
therefore B also measures C according to the units in A. 

Therefore A by multiplying B has made C. 

Since then A by multiplying itself has made B, and by 
multiplying B has made C, r I ■ • I'l ■ .^ I 

therefore C is cube. 

And, since C, D, E, F are in continued proportion, and C 
is cube, •_■(■.■ 11'*" 'J"^! • '■':! ' 

therefore F is also cube. fvui. 33] 

But it was also proved square ; 

therefore the seventh from the unit is both cube and square. 

Similarly we can prove that all the numbers which leave 
out five are also both cube and square. 

Q. E. D. 

If t, a, og, o,, ... be a geometrical progression, then a,, a^, a„ ... are 
^uaies; 

a,, a,, a,, ... are cubes ; 
tf„ a„, ... are both squares and cubes. ■'"■ '* ' ' ' '"' 

Since 1 : a=^a : at, 

a, = a". 

And. since a,, a,, a^ are in geometrical progression aitd a, (= a*) is a square^ 
a, is a square. [viii. 31] 

Similarly a,, a^, ... are squares. 

Next, I : a = o, ; uj ., ..•; 

whence aj=a', a cube number. "'^ , ' ~" 

And, since Oj, anOi, a, are in geometrical progression, and a, is a cube, 
a, is a cube. [viii. 33] 



39» BOOK IX [tx. 8, 9 

Similarly a„ a,,, ... are cubes. ' Vi t 

Clearly then a,, Ou, o„, ... are both squares and cubes. 
The whole result is of course obvious if the geometrical progression is 
written, with our notation, as 

I, a, a', a", a*, ... e". 



Proposition 9. 

T/as many numbers as we please beginning from an unit be 
in continued proportion, and the number after the unit be square, 
all the rest will also be square. And if Ihs number after the 
unit be cube, all the rest will also be cube. 

Let there be as many numbers as we please, A, B, C, D, 
E, F, beginning from an unit and in con- 
tinued proportion, and let A, the number a 

after the unit, be square ; ^ 

I say that all the rest will also be square. ^ 

Now it has been proved that B, the £ 

third from the unit, is square, as are also f 

all those which leave out one ; [ix. 8] 

I say that all the rest are also square. 



For, since A, B, C are in continued proportion, 
and A is square, , 

therefore C is also square. [vm. n] 

Again, since B, C, D are in continued proportion, 
and B is square, 
D is also square. [vm. 22] 

Similarly we can prove that all the rest are also square. 

Next, let A be cube ; 
I say that all the rest are also cube. 

Now it has been proved that C, the fourth from the unit, 
is cube, as also are all those which leave out two ; [ix, 8] 

I say that all the rest are also cube. 

For, since, as the unit is to y^, so is /^ to B, 
therefore the unit measures A the same number nf times as A 
measures B. 

But the unit measures A according to the units in it ; 
therefore A also measures B according to the units in itself; 
therefore A by multiplying itself has made B, 



IX. 9, lo] PROPOSITIONS 8—10 393 

And ^ is cube. - ' 

But, if a cube number by multiplying itself make some 
number, the product is cube. [ix. 3] 

Therefore B is also cube. 

And, since the four numbers A, B, C, D are in continued 
proportion, 

and >^ is cube, . ., . - , ,^ :.. ... 
D also is cube. , , ■ mii [v'm. 23] 

For the same reason 
E is also cube, and similarly all the rest are cube. 

., Q. E. D. 

If I, a*, tf,, a,, a^, ... are in geometrical progression, t>,, a,, a^, ... are all 
squares; 

and, if I, a", di, Ott ^t ■•• ^c in geomt^Crical progression, o^, a,, ... are all cubes, 
(i) By IX. 8, IT,, 1I4, a,, ... are all squares. 
And, a', ii„ a, being in geometrical progression, and o' being a square, 

a^'viA square. [viii, 23] 

For the same reason a,, Ot, ... are at) squares. 
(1) By IX. 8, 17,, a,, a„ ... are all cubes. . 

Now I : o* = «* : «j. 

Therefore <»i = (J* . o*, which is a cube, by ix. 3. 

And, <^, 011 "■> <it being in geometrical progression, and a* being cube, 

a, is cube. [^"i- '3] 

Similarly we prove that o, is cube, and so on. 
The results are of course obvious in our notation, the series being 
(i) I, a', a*, a*. ... o* 

Proposition id. 

l/as many numbers as we piease beginning from an unit be 
in continued proportion, and the number after the unit be not 
square, neither will any other be square except the third from 
the unit and all those which leave out one. And, if the number 
after the unit be not cube, neither will any other be cube except 
the fourth from the unit and all those which leave out two. 

Let there be as many numbers as we please. A, B, C, D, 
E, F, beginning from an unit and in continued proportion, 
and let A^ the number after the unit, not be square ; 



3m BOOK IX [ix. lo 

I say that neither will any other be square except the third 

from the unit <and those which 

leave out one > . a ' 

For, if possible, let C be square. b 

But B is also square ; [«. 8] 2 

[therefore B, C have to one another ° 
the ratio which a square number ^ 
has to a square number]. ^ 

And, as B is to C, so is ^ to ^ ; ' "'*" • 

therefore A, B have to one another the ratio which a square 

number has to a square number ; 

[so that Ay B are similar plane numbers]. [vin. 26, converse] 

And B is square ; 
therefore A is also square : 
which is contrary to the hypothesis. 

Therefore C is not square. 

Similarly we can prove that neither is any other of the 
numbers square except the third from the unit and those which 
leave out one 

Next, let A not be cube, 

1 say that neither will any other be cube except the fourth 
from the unit and those which leave out two. 
For, if possible, let D be cube. 

Now C is also cube ; for it is fourth from the unit. [ix. 8] 
And, as C is to /), so is ^ to C ; 

therefore B also has to C the ratio which a cube has to a cube. 

And C is cube ; 
therefore B is also cube. [vm. 15] 

And since, as the unit is to A, so is A to B, 

and the unit measures A according to the units in it, 

therefore A also measures B according to the units in itself ; 

therefore A by multiplying itself has made the cube number ^, 

But, if a number by multiplying itself make a cube number, 
it is also itself cube. [ix. 6] 

Therefore A is also cube : 

which is contrary to the hypothesis. 

Therefore D is not cube. 



IX. lo, itj PROPOSITIONS lo, II 395 

Similarly we can prove that neither is any other of the 
numbers cube except the fourth from the unit and those which 
leave out two. 

Q. E. D. 

ir I, a, a^, a,, at, ... be a geometrical progression, then (i), if a is nut a 
square, none of the terms will be square except a^, Of, n,>, ...; 
and (a), if a is not a cube, none of the terms will be cube except «j, a„ a„, 

With reference to the first part of the proof, viz. that which proves that, if 
a, is a square, a must be a square, Heiberg remarks that the words which 
I have bracketed are perhaps spurious; for it is easier to use vui. 24 than 
the converst of viii, 26, and a use of vui. 24 would correspond better to the 
use of VIII. 15 in the second part relating to cubes. I agree in this view and 
have bracketed the words accordingly. {See however note, p, 383, on 
converses of viii, 26, 27 given by Heron.) If this change be made, the 
proof runs as follows. 

(i) If possible, let 03 be square. ' ' 

Now Q;, ; dj = a : fij. 

But ^, is a square. [ix. 8] 

Therefore a is to a, in the ratio of a square to a square. 

And <(j is square ; 

therefore a is square [viii. 24] : which is impossible. 

(3) If possible, let «< be a cube. 

Now a, ; a, = a, : a,. 

And a, is a cube. [ix. 8] 

Therefore a, is to «, in the ratio of a cube to a cube. 

And a, is a cube : 
therefore a, is a cube. [viu. 35] 

But, since i : a = a •.oj, 

tij = a'. 

And, since a* is a cube, 

a must be a cube [ix. 6] : which is impossible. 

The propositions viii. 24, 35 are here not quoted in their exact form in 
(hat the _firsi and uccnd squares, or cubes, change places. But there is no 
difficulty, since the method by which the theorems are proved shows that 
either inference is equally correct 



Proposition ii. 

If as many numbers as we please beginning from^ an unit be 
in continued proportion, the less measures the greater according 
to some one of the numbers which have place among the propor- 
tional numbers. 



30 , BOOK IX [ix. ti 

•In Let there be as many numbers as we please, B, C, D, E, 
beginning from the unit A and in con- 
tinued proportion ; ^ 

I say that B, the least of the numbers B, a 

C, D, E, measures E according to some c- 

one of the numbers C, D. o 

For since, as the unit A is to B, so ^ 

is D to E, 

therefore the unit A measures the number B the same number 

of times as D measures E \ ^ V'i"., . It. T , 

therefore, alternately, the unit A measures D the same number 

of times as B measures E. [vii. 15] 

But the unit A measures D according to the units in it ; 
therefore B also measures E according to the units in D ; 
so that B the less measures E the greater according to some 
number of those which have place among the proportional 
numbers. — 

PoRiSM. And it is manifest that, whatever place the 
measuring number has, reckoned from the unit, the same 
place also has the number according to which it measures, 
reckoned from the number measured, in the direction of the 
number before it. — 

Q. E. D. 

The proposition and the porism together assert that, if t, a, a,, ... a„ be a 
geometrical progression, a, measures a» and gives the quotient o,., {r < it). 
Eudid only proves that a^ = a.a^.„ as follows. 

Therefore i measures a the same number of times as «„., measures a,. 
Hence i measures a„., the same number of times as a measures a, ; 

. . [vn. 15} 

that IS, a, = a,ai,_,. 

We can supply the proof of the porism as follovrs. 
I : a = Br : a,^,, 
(J : n, = «,+! : a^+j, 

whence, tx aequaii, 

I : a,^, = fl, : a,. [vil. 14] 

It follows, by the same argument as before, that 

With our notation, we have the theorem of indices that 



IX. i2l PROPOSITIONS II, I* ^ 

Proposition 12. 

If as many numbers as we please beginning from an unit be 
tn conlitmed proportion, by however many prime numbers (he 
lasi is measured, the next to the unit will also be measured by 
ike same. 

Let there be as many numbers as we please, A, B, C, D, 
beginning from an unit, and in continued proportion ; 
I say that, by however many prime numbers D is measured, 
A will also be measured by the same. 



A P- " ,. ', 

B Q 

• O 

For let /? be measured by any prime number £ ; 
I say that £ measures A. 

I" For suppose it does not; " - ' 

now £ is prime, and any prime number is prime to any which 
it does not measure ; [vn. 29] 

therefore £, A are prime to one another. 

And, since E measures D, let it measure it according to F, 
therefore £ by multiplying F has made D. 

Again, since A measures D according to the units in C, 

[ix, 1 1 and For,] 
therefore A by multiplying C has made D. 

But, further, £ has also by multiplying F made D ; 
therefore the product of ^, C is equal to the product of £, F. 

Therefore, as A is to B, so'is F to C. [vii> 19] 

But .(4, .£ are prime, .< j :"i : > 

primes are also least, [vn. 21] 

and the least measure those which have the same ratio the 
same number of times, the antecedent the antecedent and the 
consequent the consequent ; ., ^ [vn. 20] 

therefore E measures C. 
., Let it measure it according to G; - ' • 

therefore £ by multiplying G has made C. 

But, further, by the theorem before this, 
A has also by multiplying B made C. [ix. 1 1 and Por.] 



^ c> . BOOK IX- [ix. 13 

Therefore the product of A, B is equal to the product of 

Therefore, as -,4 is to £", so is ff to J?. ''^ " [vil 19] 

But A, E are prime, 
primes are also least, [vii. ai] 

and the least numbers measure those which have the same 
ratio with them the same number of times, the antecedent the 
antecedent and the consequent the consequent : L^ii. 10] 

therefore H measures B. 

Let it measure it accordinp; \o H \ 

therefore E by multiplying H has made B, 

But further A has also by multiplyiner itself made B ; 

[.X.81 

therefore the product of E, H is equal to the square on A. 

Therefore, as £" is to ^, so is ^ to //". [vii. 19] 

But A, E are prime, ..; •> I 

primes are also least, [vn. zi] 

and the least measure those which have the same ratio the 
same number of times, the antecedent the antecedent and the 
consequent the consequent ; [vn. so] 

therefore E measures A, as antecedent antecedent 

But, again, it also does not measure it : , 

which is impossible^ -iii.. • '...if! . /', 

Therefore E, A are not prime to one another. ^ ^ 
Therefore they are composite to one another. '' ''' " ' 
But numbers composite to one another are measured by 

some number, \^i\. Def. 14] 

And, since E ij by hypothesis prime, 

and the prime is not measured by any number other than itself, 

therefore E measures A, E, 

so that E measures A. > • ^ . . ' ,• 

[But it also measures /? ; ■"*-' ■ . i 1. iv- 

therefore E measures A, Z?.] 

Similarly we can prove that, by however many prime 

numbers D is measured, A will also be measured by the same. 

Q. E. D, 

If I, a, 0,, ... a^hea. geometrical progression, and a. be measured by any 
prime number >, a will a.tso be measured by /. 



IX. I*, 13] PROPOSITIONS 12, 13 399 

For, if possible, suppose that p does not measure a J then,/ being prime, 

/, a are prime to one another. . ^ [vii. 29] 

Suppose a^^m.p. ,.".,. .,., 

Now • o, = a.a,.,. ' ' [ix. li] 

Therefore o . o,., = « . /, 

and a : p = m : a^-i. • ' > [vil. 19] 

Hence, n,/ being prime to one another, 

p measures ii„_i. [vil. zo, ir] 

By a reptetition of the same process, we can prove that p measures a„.j 
and therefore o,_„ and so on, and finally that p measures a. 

But, by hypothesis, / does not measure a : which is impossible. 

Hence p, a are not prime to one another : 
therefore they have some common factor. [vu. Def. 14] 

Butp is the only number which measures p; 
therefore/ measures a. 

Heiberg remarks that, as, in the iii0iirK, Euclid sets himself to prove that 
E measures j4, the words bracketed above are unnecessary and therefore 
perhaps interpoiatedi ■ » . • 



Proposition 13, 

//as many numbers as we piease beginning from an unit be 
in continued proportion, and the number after the unit be prime, 
the greatest will not be measured by any except those which, have 
a place among the proportional numbers. 

Let there be as many numbers as we please. A, B, C, D, 
beginning from an unit and in continued proportion, and let A, 
the number after the unit, be prime ; 

I say that Z?, the greatest of them, will not be measured by any 
other number except A, B, C. 

K e '••'•'^■^ 

B F ■ i' ' "■^■•\ 

C Q. 

O H 

For, if possible, let it be measured by E, and let E nqt be 
the same with any of the numbers A, B, C. 

It is then manifest that E is not prime. ' ' "' 

For, if £ is prime and measures D, 
it will also measure A [ix. 12], which is prime, though it is not 
the same with it : 
which is impossible. -rvrii .- 1. • . '. 



4m BOOK. IX [ix. 13 

Therefore -£" is not prime. ■-:■-■ .m.^. ■„.. . .1. 

Therefore it is composite. 

But any composite number is measured by some prime 
number ; [vii. 31] 

therefore B is measured by some prime number. 

I say next that it will not be measured by any other prime 
except A. 

For, if £ is measured by another, •■ . . 
and £ measures D, • , 1, 

that other will also measure D\ 

so that it will also measure A [ix. n], which is prime, though 
it is not the same with it : 

which is impossible. 

Therefore A measures E. 

And, since E measures D, let it measure it according to F. 

I say that F is not the same with any of the numbers 
A, B, C. 

For, if F is the same with one of the numbers A, B,C, 
and measures D according to A, 

therefore one of the numbers A,B,C also measures D according 
to E. 

But one of the numbers A, B, C measures D according to 
some one of the numbers A, B, C; [ix. 11] 

therefore E is also the same with one of the numbers A, B,C: 
which is contrary to the hypothesis. 

Therefore F is not the same as any one of the numbers 
A. B, C. 

Similarly we can prove that F is measured by A, by 
proving again that Fi& not prime. 

For, if it is, and measures Z?, 

it will also measure A [ix, i *], which is prime, though It is not 

the same with it : 

which is impossible ; 

therefore F is not prime. 

Therefore it is composite. 

But any composite number is measured by some prime 
number ; [va. 31 j 

therefore F is measured by some prime number. 



IX. 13] PROPOSITION 13 401 

I say next that it will not be measured by any other prime 
except A. 

For, if any other prime number measures F, 
and F measures D, 

that other will also measure D ; 

so that it will also measure A [ix. u], which I's prime, though it 
is not the same with it : 

which is impossible. , .,^,, ,,u . 

Therefore A measures F. 

And, since E measures D according to ^^ ., .^^ 
therefore E by multiplying F has made D. ■ '• ' 

But, further, A has also by multiplying C made D; [ix, n] 

therefore the product of A, C is equal to the product of F, F. 

Therefore, proportionally, as A is to E, so is F to C. 

[vii. 19] 
But A measures E ; . 

therefore F also measures C, 

Let it measure it according to G. "' ' 

Similarly, then, we can prove that G is not the same with 
any of the numbers A, B, and that it is measured by A. 
And, since F measures C according to G 

therefore F by multiplying G has made C. 

But, further, A has also by multiplying B made C \ [ix. it] 
therefore the product of A, B is equal to the product of F, G. 

Therefore, proportionally, as A is to F, so is G to B. 

[v"' '9] 
But y4 measures /^; " ' ' ■. ', ,, 

therefore G^ also measures ^. ' •"•' 

Let it measure it according to H. 

Similarly then we can prove that H is not the same 
with A. 

And, since G measures B according to H, 

therefore G by multiplying H has made B, 

But further A has also by multiplying itself made B ; 

[ix. 8] 

therefore the product of H, G is equal to the square on A. 

Therefore, as // is to ^, so is v4 to 6^. [vil 19] 



jfjaa BOOK IX [ix. 13, 14 

But A measures (7; ' ' ■ > ■ . ■ . ■,. 1 

therefore H also measures A, which is prime, though it is not 
the same with it : 
which is absurd. 

Therefore D the greatest will not be measured by any 
other number except A, B, C. 

• Q. E. D. 

If I, a, a,, ... a„ b« a geometrical progression, and if a is prime, a. will not 
be measured by any numbers except the preceding terms of the series. 

ir possible, let a^ be measured by by a number different from all the 
preceding terms. 

Now b cannot be prime, for, if it were, it would measure a. [ix. 12] 

Therefore b is composite, and hence will be measured by somt prime 
number [vii. 31], say p. 

Thus p must measure a„ and therefore a [ix. i i] ; so that / cannot be 
different from a, and b is not measured by any prime number except a. 

Suppose that a^~b . r. 

Now f cannot be identical with any of the terms a, a,, ... a^-ti for, if it 
were, 6 would bt identical with another of them: [ix, 11] 

which is contrary to the hyfwthesis. 

We car non- prove (just as for b) that c cannot be prime and cannot be 
measured by any prime number except a. 

Since l>.e-a^-a . a^.i, [ix. Ii] 

a:i = e: a,_, , 
whence, since a measures i, 

IT measures a^-i. 

Let a^_, = t .d. 

We now prove in the same way that d is not identical with any of the terms 
0, a■^, ... a,_i, is not prime, and is not measured by any prime except a, and 
also that 

d measures a,.|. 

Proceeding in this way, we get a last factor, say k, which measures a 
though different from it : 
which is absurd, since a is prime. 

Thus the original supposition that a„ can be measured by a number S 
different from all the terms a, a^, ... a„_, must be incorrect. 

Therefore etc. 

Proposition 14. 

//a number be (he least that is measured by prime numbers, 
it will not be measured by any other prime number except those 
originally measuring it. 

For let the number A be the least that is measured by the 
prime numbers B, C, D; 



IX. 14] PROPOSITIONS 13, 14 403 

I say that A v/iW not be measured by any other prime number 
except B, C, Z>. 

For, if possible, let it be measured by the prime number 
£, and let £ not be the same with any one of the numbers 
B. C, D. 



A 


B 






F 



Now, since E measures A, let It measure it according 
to F\ 

therefore E by multiplying F has made A. 

And A is measured by the prime numbers B, C, D. 

But, if two numbers by multiplying one another make some 
number, and any prime number measure the product, it will 
also measure one of the original numbers ; [vii. 30] 

therefore B, C, D will measure one of the numbers E, F. 

Now they will not measure E ; 

for E is prime and not the same with any one of the numbers 
B, C D. 

Therefore they will measure F, which is less than A : 

which is impossible, for A is by hypothesis the least number 
measured by B, C, D. 

Therefore no prime number will measure A except 
B, C, D. 

Q. E. D. 

In other words, a number can be resolved into prime factors in only 
one way. 

Let a be the least number measured by each of the prime numbers 
i, t, d, ... k. 

If possible, suppose that a has a prime factor/ different from />, c, d, ... k. 

Let a-p.m. 

Now /i,(,d, ... a, measuring «, must measure one of the two factors/, m. 

[vii. 30] 
rhey do not, by hypothesis, measure p ; 

therefore they must measure iw, a number less than a: "•.)""•";.(-'. 

which is contrary to the hypothesis. 

Therefore a has no prime factors except i, (, d, ... k. 



404 *» - BOOK IX I [ix. IS 

Proposition 15, 

// three numbers in continued proportion be the least of 
those which have the same ratio with them, any two whatever 
added together will he prime to the remaining number. 

Let A, B, C, three numbers in continued proportion, be 
the least of those which have the same 

ratio with them ; * b ■ 

I say that any two of the numbers c 

A, B, C whatever added together are 0— J — f 

prime to the remainingnumber, namely . 

A, B to C\ B, Cto A ; and further A, C to B. 

For let two numbers DB, EF, the least of those which 
have the same ratio with A, B, C, be taken. [vni. 1] 

It is then manifest that I>B by multiplying itself has made 
A, and by multiplying BB has made B, and, further. BB by 
multiplying itself has made C. [vin. a] 

Now, since DB, BBare least, 
they are prime to one another. [vii. «] 

But, if two numbers be prime to one another, 
their sum is also prime to each ; [vii. a8] 

therefore DB is also prime to each of the. numbers DB, BB. 

But further D£ is also prime to BB ; ^ 

therefore DB, DB are prime to BB. 

But, if two numbers be prime to any number, 
their product is also prime to the other ; [vii. 84] 

so that the product of BD, DB is prime to BF\ 
hence the product of FD, DB is also prime to the square 
on BF. [vii. »j] 

But the product of BD, DE is the square on DB together 
with the product of DB, BF; [n. 3] 

therefore the square on DB together with the product of DB, 
BF is prime to the square on BF. 

And the square on DB is A, 
the product of DE, BB is B, .,...., 
and the square on BF is C; 
therefore A, B added together are prime to C. 



IX. is] proposition 15 405 

Similarly we can prove that B, C added together are 
prime to A. 

I say next that A, C added together are also prime to B. 
For, since DF is prime to each of the numbers DE, EF, 

the square on DF is also prime to the product of DE, EF. 

[vii. 24, js] 

But the squares on DE, EF together with twice the pro- 
duct of DE, EFa.K equal to the square on DF; [11. 4] 
therefore the squares on DE, EF together with twice the 
product of DE, EF are prime to the product of DE, EF. 

Separando, the squares on DE, EF together with once 
the product of DE, EF^e prime to the product of DE, EF. 

Therefore, separando again, the squares on DE, EF are 
prime to the product oi DE, EF. '' "'"'<• "- 

And the square on Z?^ is v4, :V . •.' . . 

the product of DE, EF is B, 
and the square on EF is C. : " 

Therefore A, C added together are prime to B. 

Q. E, D. 

If a, 6, c he a. geometrical progression in the least terms which have a 
given common ratio, (i + f), (c+ 3), (a + *) are respectively prime to a, d, e. 

Let a : ^ be the cominon ratio in its lowest terms, so that the geometrical 
progression is 

a\ a/9, f?. ' ' [VHI. »J 

Now, a, fi being prime to one another, ' ' 

a-*-/} is prime to both a and j8, " [vii. *8] 

Therefore (0 + ^)1 <> ^t* t^th prime to p. 

Hence (a + ^ a is prime to j3, [vit. 24] 

and therefore to ^; [vii. 25] 

Le- a' + afi is prime to ^, 

or ' a + i is prime to ^. " 

Similarly, ' ' c^ + jS* is prime to a', * * 

or * + f is prime to o. . ' ■' **''■ 

Lastly, a ■«■ ^ being prime to both a and j9, 

(a + ^)' is prime to a/9, [vii. 24, 25] 

or o' + /S* + Mj9 is prime to afi : 

whence a' + yS" is prime to aj8. 

The latter inference, made in two steps, may be proved by reducHo ad 
absurdum as Commandinus proves it 

If a* + ^ is not prime to o/S, let x measure them ; 
therefore x measures a* + ^ ■«- lafi as well as o^ ; 

hence a* + ^ ■«- 2 aj3 and aj3 are not prime to one another, which is contrary 
to the hypothesis. 



4o6 BOOK IX [ix. i6, i? 

Proposition i6. 

If two numbers be prime to one another, the second will not 
be to any other number as the first is to the second. 

For let the two numbers A, Bhe. prime to one another ; 
I say that B is not to any other number as 
A is to B. A 

For, if possible, as A is to B, so let B be b 

to C. o 

Now A, B are prime, 
primes are also least, fvii. zi] 

and the least numbers measure those which have the same 
ratio the same number of times, the antecedent the antecedent 
and the consequent the consequent ; [vii. 20] 

therefore A measures B as antecedent antecedent. 

But it also measures itself; 
therefore A measures A, B which are prime to one another : 
which is absurd. 

Therefore B will not be to Q as A is to B. 

If a, h are prime to one another, they can have no integral third 
proportional. 
^^ If possible, let a : 6 = i ; x. 

Therefore [vii. zo, 21] a measures i, and a. b have the commorv measure, 
d, which is contrary to the hypothesis. ^ ,.,.,,. a , . 

.• 'r-r Proposition 17. "» 

■ - If there be as many numbers as we please in continued 
proportion, and the extremes of them be prime to one another, 
the last will not be to any other number as the first to the 
second. 

For let there be as many numbers as we please, A,B,C,D, 
in continued proportion, 
and let the extremes of them, A, 
D, be prime to one another ; ^ 

1 say that D is not to any other ^ ^ 

number as A is to B, 

For, if possible, as A is to B, so let D he to £ ; 
therefore, alternately, as y^ is to Z?, so is .5 to .£", [vii. 13] 



: J-' 


- '^i 


-.■j '- 


,^^ii '- 


'\ 13(1 til 


' ■ • 1 


I I'ur'- 1 


• ?o; 1 




\ti 



IX, 17, 18] PROPOSITIONS 16—18 407. 

But A, D are prime, ~ ' ' ' • 1 

primes are also least, if.' i" i-* yj [vii.it] 

and the least numbers measure those which have the same 
ratio the same number of times, the antecedent the antecedent 
and the consequent the consequent. [vn. 20] 

Therefore A measures ^. 

And, as A is to B, so is B to C. 

Therefore B also measures C \ 
so that A also measures C. 

And since, as B is to C, so is C to D, 
and ^ measures C, 
therefore C also measures D. , , „ 

But y4 measured C ; 
so that A also measures D. , , 

But it also measures itself; 
therefore A measures A, D which are prime to one another : 
which is impossible. 

Therefore Z? will not be to any other number as A is to B. 

Ifd, 1(9, d„ ... Sh be a geometrical prc^iression, and a, a. are prime to one 
another, then a, a„ a„ can have no integral fourth proportional. 

For, if possible, let a ; a, = a, : jf, 

Therefore a : a^ = a^\ x, 

and hence [vn. 20, 21] a measures at. 

Therefore a, measures a,, [vii. Def. 20] 

and hence a measures a,, and therefore also ultimately a,. 

Thus a, a^ are both measured by a : which is contrary to the hypothesis. 

Proposition 18. 

Given two numbers, to investigate whether it is possible to 
find a third proportional to them. 

Let A, B be the given two numbers, and let it be required 
to investigate whether it is possible to find a third proportional 
to them. 

Now A, B are either prime to one another or not. 

And, if they are prime to one another, it has been proved 
that it is impossible to find a third proportional to them. 

[tx. 16] 



4«8 >< BOOK IX <y.-' [ix. 18 

Next, let A, B not be prime to one another, 
and let B by multiplying itself make C. 

Then A either measures C or does not measure it. 



A- 
B- 



First, let it measure it according to D ; 
therefore A by multiplying D has made C. 

But, further, B has also by multiplying itself made C ; 
therefore the product of ^, D is equal to the square on B, 

Therefore, as ^ is to B, so is /? to Z? ; [vii. 19] 

therefore a third proportional number D has been found to 
A,B. 

Next, let A not measure C ; 
I say that it is impossible to find a third proportional number 
to A, B. ^ 

For, if possible, let /?, such third proportional, have been 
found. 

Therefore the product of A, D is equal to the square on B. 

But the square on ^ is C; 
therefore the product of A, D is equal to C. 

Hence A by multiplying D has made C \ 
therefore A measures C according to D. 

But, by hypothesis, it also does not measure it : 
which is absurd. 

Therefore It is not possible to find a third proportional 
number xja A^ B when A does not measure C. q. e. d. 

Given two numbers a, 6, to find the condition that they may have an 
int^ral third proportional, 

(t) a, i must not be prime to one another. [ix. 16] 

(a) a must measure S'. 

For, if a, ^, f be in continued proportion. 

Therefore a measures ^. 
Condition (i) is included in condition (1) since, if l^ = ma, a and S cannot 
be prime to one another. 

The result is of course easily seen if the three terras in continued 
proportion be written 

"' "a' Hi}- 



IX. rg] PROPOSITIONS i8, 19 j|«9 



Proposition 19. ^''i ■•'• " ' '■■■ ' 

Given three numben, to investigate when it is possible to 
find a fourth proportional to tkem. 

Let A, B,C\x:^ the given three numbers, and let it be 
required to investigate when it is ^^ _ 

possible to find a fourth proportional ^ , 

to them. g .,1' 

Now either they are not in con- p , , 

tinued proportion, and the extremes ^ 

of them are prime to one another ; 

or they are in continued proportion, and the extremes of them 

are not prime to one another ; 

or they are not in continued proportion, nor are the extremes 

of them prime to one another ; 

or they are in continued proportion, and the extremes of them 

are prime to one another. 

If then A, B, C are in continued proportion, and the 
extremes of them A, C are prime to one another, 
it has been proved that it is impossible to find a fourth pro- 
portional number to them. [ix. 17] 

tNext, let A, B, C not be in continued proportion, the 
extremes being again prime to one another ; 
I say that in this case also it is impossible to find a fourth 
proportional to them. 

For, if possible, let D have been found, so that, 
as y^ is to ^, so is C to D, 
and let it be contrived that, as ^ is to C, so is Z> to E. 

Now, since, as A is to B, so is C to D, 
and, as ^ is to C, so is Z? to .£", 
therefore, ex aequali, as ^4 is to C, so is C to B. [vii. 14] 

But A, C are prime, 
primes are also least, [vit. li] 

and the least numbers measure those which have the same 
ratio, the antecedent the antecedent and the consequent the 
consequent [vn, ao] 

Therefore A measures C as antecedent antecedent. 



4W ^i BOOK IX ■• [ix. 19 

But it also measures itself ; 
therefore A measures A, C which are prime to one another : 
which is impossible. 

Therefore it is not possible to find a fourth proportional 

to A. B, C.f 

Next, \et A, B, C be again in continued proportion, 
but let A, C not be prime to one another. 

I say that it is possible to find a fourth proportional to 
them. 

For let B by multiplying C make D ; 
therefore A either measures D or does not measure it. 

First, let it measure it according to E ; 
therefore A by multiplying E has made D. 

But, further, B has also by multiplying C made D ; 
therefore the product of A, E is equal to the product of 
B, C; 

therefore, proportionally, as A is to B, so is C to ^ ; [vii. 19] 
therefore E has been found a fourth proportional to A, B, C. 

Next, let A not measure D ; 
I say that it Is impossible to find a fourth proportional number 
to A, B, C. 

For, if possible, let E have been found ; 

therefore the product of A, E is equal to the product of^, C, 

[vii. 19] 

But the product if B, C is D ; 
therefore the product of A, E is also equal to D. 

Therefore A by multiplying E has made D ; 
therefore A measures D according to E, 
so that A measures D. 

But it also does not measure it : 
which is absurd. 

Therefore it is not possible to find a fourth proportional 
number to A, B, C when A does not measure D. 

Next, let A, B, C not he in continued proportion, nor the 
extremes prime to one another. 

And let B by multiplying C make D. . 1 • 

Similarly then it can be proved that, if A measures D, 
it is possible to find a fourth proportional to them, but, if it 
does not measure it, impossible, Q. e. d. 



IX. ig] PROPOSITION ig 411 

Given three numbers a, b, e, to find the condition that they may have an 
integral fourth proportional. 

The Greek text of part of this proposition is hopelessly corrupt. Accord- 
ing to it Euclid takes four cases, 

(i) a,b,e not in continued proportion, and a, c prime to one another. 

(2) a, ^, c in continued proportion, and a, c not prime to one another. 

(3) o, *i <^ not in continued proportion, and a, t not prime to one another. 

(4) a,b,c in continued proportion, and a, c prime to one another. 

(4) is the case dealt with in ix. 17, where it is shown that on hypothesis 
{4) a fourth proportional cannot be found. 

The text now takes case (i) and asserts that a fourth proportional cannot 
be found in this case either. We have only to think of 4, 6, 9 in order to see 
that there is something wrong here. The supposed proof is also wrong. If 
possible, says the text, let rf Ije a fourth proportional to a, b, c, and lei e 
be taken such that 

b : c - d : e. 

Then, ex a^uali, n '. c-c ; e, 

whence a measures c : [vii, so, Ji] 

which is impossible, since o, c are prime to one another. 

But this does not prove that a fourth proportional d cannot be found ; it 
only proves that, if if is a fourth proportional, no integer e can be found to 
satisfy the equation , - .^ 

b : e-d ; e. 

Indeed it is obvious from ix. 16 that in the equation f- ■ •• . 

a : € = t \e " 

t cannot be integral. > . 

The cases {%) and (3) are correctly given, the first in full, and the other as 
a case to be proved "similarly" to it. 

These two cases really give all that is necessary. 

Let the product be be taken. 

Then, if a measures be, suppose bc = ad; 
therefore a : b = c : d, 

and ^ is a fourth proportional. 

But, if a does not measure be, no fourth proportional can be found. 
For, if X were a fourth proportional, ax would be equal to />c, and a would 
measure be. 

The sufficient condition in any case for the possibility of finding a fourth 
proportional to a, b, e is that a should measure be. 

Theon appears to have corrected the proof by leaving out the incorrect 
portion which I have included between daggers and the last case {3) dealt 
with in the last lines. Also, in accordance with this arrangement, he does not 
distinguish four cases at the beginning but only two. " Either A, B, C are 
in continued proportion and the extremes of them A, C are prime to one 
another; or not," Then, instead of introducing case (2) by the words 
"Next let A, B, C.to find a fourth proportional to them," immediately 
following the second dagger above, Theon merely says " But^ if not," [i.e. 
if it is not the case that a, b, ^ are in g.p. and a, ^ prime to one another] "let 
B by multiplying C make D," and so on. 



4H BOOK IX ! [ix. 19, 10 

August adopts Theon's fonn of the proof. Heibeig does not feel able to 
do this, in view of the superiority of the authority for the text as given above 
(P) ; he therefore retains the latter vrithout any attempt to emend it. 



'■ "•'"' Proposition 20. ' •'•""•'•' ^-" • 'M 

Prime numbers are more than any assigned muUitvde of 
prime numbers. iutt t ji .-i -irrw )>. 

Let ^, jff, C be the assigned prime numbers ; 

I say that there are more > i^' 

prime numbers than A, B, C. a— •''■ ' ' 

For let the least number ^ Q 

measured by ^, .5, C be c 

taken, e ^F 

and let it be DE ; 

let the unit i?/^ be added to ZJ.fi'. 

Then EF is either prime or not. ' '• 

First, let it be prime ; 
then the prime numbers A, B, C, EEhave been found which 
are more than A, B, C. 

Next, let ^"^ not be prime; ; '-'a > • -i- 

therefore it is measured by some prime number [vir. 31] 

Let it be measured by the prime number G. 

I say that G is not the same with any of the numbers 
A,B,C. 

For, if possible, let it be so. 

Now A, B, C measure DE ; ■" '• 

therefore G also will measure DE. '^\ ^, ',' ",' |' 

But it also measures EE. 

Therefore G, being a number, will measure the remainder, 
the unit i?^: .., . , , . , 

which is absurd. 

Therefore G is not the same with any one of the numbers 
A,B,C. . I , 

And by hypothesis it is prime. 
• Therefore the prime numbers A, B, C, G have been found 
which are more than the assigned multitude of A, B, C. 

Q. E. D. 



IX. 20-I*] PROPOSITIONS 19—2* 413 

We have here the important proposition that the number of prime numiert 
ts infinite. 

The proof will be seen to be the same as that given in our algebraical 
text-books. Let a, ^, e, ..■ Ji be any prime numbers. 

Take the product aie ... k and add unity. 

Then {ai( ... i + i) is either a prime number or not a prime number. 

(i) If it /r, we have added another prime number to those given. 

(2) If it is not, it must be measured by some prime number [vii. 31], say/. 

Now/ cannot be identical with any of the prime numbers a, b,e, ... k. 

For, if it [s, it will divide abc ... k. 
Therefore, since it divides l^abc...'k.^ i) also, it will measure the difference, 
or unity : 
which is impossible. 

Therefore in any case we have obtained one fresh prime number. 

And the process can be carried on to any extent. 

Proposition 21. 

If as many even numbers as we please be added together, 
tke whole is even. 

For let as many even numbers as we please, AB, BC, CD, 
DE, be added together ; 

I say that the whole AE * b c o e 

is even. 

For, since each of the numbers AB, BC, CD, DE is even, 
it has a half part ,; [vii, Def. 6] 

so that the whole AE also has a half part. 

But an even number is that which is divisible into two 
equal parts ; [«A] 

therefore AE is even, 

Q. E. D. 

In this and the following propositions up to ix. 34 inclusive we have a 
number of theorems about odd, even, "even-times even" and "even- times 
odd" numbers respectively. They are all simple and require no explanation 
in order to enable them to be followed easily. 

Proposition 22, 

If as many odd numbers as we please be added together, and 
their multitude be even, the whole will be even. 

For let as many odd numbers as we please, AB, BC, CD, 
DE, even in multitude, be added together ; , 

I say that the whole AE is even. 



414 " BOOK IX [ix. aa-H 

For, since each of the numbers AB, BC, CD, DE is odd, 
if an unit be subtracted from each, each of the remainders will 
be even ; [vn, ]>ef. 7] 

so that the sum of them will be even. > [ix. 31] 

* ^ p D E 



But the multitude of the units is also even. 

Therefore the whole AE is also even. [ix. ai] 



Q. E, D. 



Proposition 23. 

If as many odd numders as we please be added together, 
and their multitude be odd, the whole will also be odd. 

For let as many odd numbers as we please, AB, BC, CD, 
the multitude of which is odd, 

be added tc^ether ; a b c e d 

I say that the whole AD is ' *" " 

also odd. 

Let the unit DE be suht»^cted from CD ; 
therefore the remainder CE is even. 

But CA is also even ; 
therefore the whole AE is also even. 

And DE is an unit. 

Therefore AD is odd, 

3. LiteniUj' '* let there ^^c as many numbers u we please, of which /tf the muLtUude it 
odd." Thb forai, natural in Greek, is awkward in English, 



Proposition 24. 

If from an even number an even number be subtracted, the 
remainder will be even. 

For from the even number AB let the even number BC 
be subtracted : 
1 say that the remainder CA is even. a 9 a 

For, since AB is even, it has a half 
part, [vn. I>ef. 6] 





[vii 


1. Def. 


7] 






[■X. 


..] 






t.x. 


"] 




[v.> 


:. Def. 


71 


Q. 


, E. 


D. 





IX. 34-a6] PROPOSITIONS aa— a6 415 

For the same reason BC also has a half part ; 

so that the remainder [CA also has a half part, and] AC \i 
therefore even. 

• , • : . Q. E. D, 



Proposition 25. 

1/ from an even number an odd number be subtracted, the 
remainder will be odd. 

For from the even number AB let the odd number BC be 
subtracted ; 

I say that the remainder CA is odd. /k c d b 

For let the unit CD be sub- 
tracted from BC ; 

therefore DB is even. • [vn. Def. 7] 

But AB is also even ; 
therefore the remainder AD is also even. [ix. 84] 

And CD is an unit ; 
therefore CA is odd. ' [vn. Def. 7] 



I' 



Q. E. D, 



Proposition 26. 

If from an odd number an odd number be subtracted, the 
remainder will be even. 

For from the odd number AB let the odd number BC be 
subtracted \ 

I say that the remainder CA is even. p^ o d a 

For, since AB is odd, let the unit 
BD be subtracted ; ... 

therefore the remainder AD is even. [vii, Def. 7] 

For the same reason CD is also even ; [vn. Def. 7] 

so that the remainder CA is also even, [ix. 14] 

Q. E. D. 



4t6 o. BOOK DC :>'< [ix. 17— 39 

Proposition 27. 

If from an odd number an even number be subtracted, the 
remainder will be odd. 

For from the odd number j45 let the even number BC be 
subtracted ; 
I say that the remainder CA is odd. 



Let the unit AD be subtracted ; — ^ 



therefore DB is even. [vn. Def, 7] 

But BC is also even ; 
therefore the remainder CD is even. [ix. ^4] 

Therefore CA is odd. [vil IM, 7] 

Q. E. D. 

Proposition 28, •., ..- ■ 

If an odd number by multiplying an even number make 
some number, the product will be even. 

For let the odd number A by multiplying the even number 
B make C\ „ , . 
I say that C is even. 

For, since A by multiplying B has 

made C, 

therefore C is made up of as many numbers equal to B as 

there are units in A. [vn. Def. 15] 

And B is even ; 
therefore C is made up of even numbers. 

But, if as maoy even numbers as we please be added 
together, the whole is even. [ix. ai] 

Therefore C is even. 

Q. E. D. 

Proposition 29. 

If an odd number by multiplying an odd number muke 
some number, the product will be odd. 

For let the odd number A by multiplying the odd number 
B make C ; 
I say that C is odd. * 

For, since A by multiplying B has g 

made C, 



IX. 29—31] PROPOSITIONS »7— 31 417 

therefore C is made up of as many numbers equal to B as 
there are units in A. [vii. Def. 15] 

And each of the numbers A, B is odd ; 
therefore C is made up of odd numbers the multitude of which 
is odd. 

'''tins C is odd. ,., .r, --li-^Ki.- ' b^'ii 

' Q, E. D. 

Proposition 30. 

// an odd number measure an even number^ it wiU also 
measure Ike half of it. • ■• 

For let the odd number A measure the even number B ; 
I say that it will also measure the half 
of it. ^__ 

For, since A measures B, a 

let it measure it according to C ; o 

I say that C is not odd. 

For, if possible, let it be so. ' • ^ 

Then, since A measures B according to C, "'- " ** 
therefore A by multiplying C has made B. 

Therefore B is made up of odd numbers the multitude 
of which is odd. 

Therefore .5 is odd : ' n ;: \ ■ v. • ["c. 33] 

which is absurd, for by hypothesis it is even. ■• ••- ■ . 

Therefore C is not odd ; < : ^ ' 

therefore C is even. - •'"' ^c» .our, 

■ , r 

Thift A measures B an even number of times. 
For this reason then it also measures the half of it. 

Q. E, D. 

V 1 , , ,, 

Proposition 31. 

// an odd number be prime to any nutter, it will also be 
prime to the double of it. 

For let the odd number A be prime to any number B, 

and let C be double of B ; 

I say that A is prime to C. 

For, if they are not prime ^ 

to one another, some number ^ 
will measure them. "" 



0lt BOOK IX ■ ^ ■ [ix. 31, 3a 

Let a number measure them, and let it be Z*. 

Now -^ is odd ; 
therefore D is also odd. 

And since D which is odd measures C, 
and C is even, "''' ''■' 

therefore [ZJ] will measure the half of C also, ' [ix. 30] 

But ^ is half of C; 
therefore D measures B. ■■• 

But it also measures A ; 
therefore D measures /I, B which are prime to one another: 
which is impossible. 

Therefore A cannot but be prime to C. 

Therefore A, C are prime to one another. 

Q. £. D. 

Proposition 52. 

Each of the numbers which are continually doubled beginning 
from a dyad is even-times even only. 

For let as many numbers as we please, B, C, D, have been 
continually doubled beginning 
from the dyad A ; * — 

I say that B, C, D are even- ' 

times even only. . 

Now that each of the 
numbers B, C, D is even-times even is manifest ; for it is 
doubled from a dyad. 

I say that it is also even-times even only. 

For let an unit be set out. 

Since then as many numbers as we please beginning from 
an unit are in continued proportion, 
and the number A after the unit is prime, 
therefore D, the greatest of the numbers A, B, C, D, will not 
be measured by any other number except A, B, C. [ix. 13] 

And each of the numbers A, B, C is even ; 
therefore D is even-times even only. [vn. Def. 8] 

Similarly we can prove that each of the numbers B, C is 
even-times even only. 

Q. E. D. 



VL 3a— 34] PROPOSITIONS 31—34 419 

See the notes on vti. Deff. 8 to 11 for a discussion of the difficulties 
shown by lamblichus to be involved by the Euclidean definitions of " even- 
times even," "eveo-timesodd" and "odd-times even," 



Proposition 33. 

I/a number have its haif odd, it is even-times odd only. 

For let the number A have its half odd ; 
I say that A Is even-times odd only. 

Now that it is even-times odd is % 

manifest ; for the half of it, being odd, 

measures it an even number of times. [vil Def. 9) 

I say next that it is also even -times odd only. 

For, if A is even-times even also, 
it will be measured by an even number according to an even 
number ; [vii. Def. 8] 

so that the half of it will also be measured by an even number 
though it is odd : 
which is absurd. ^ , . ,, ,, 

Therefore A is even-times odd only, ^ , . Q. B. d. 

Propositioh 34- 

If a numier neither be one of those which are continually 
doubled from a dyad, nor have its half odd, it is both even- 
times even and even-times odd. 

For let the number A neither be one of those doubled 
from a dyad, nor have its half odd ■; 
I say that A is both even-times even j. 

and even-times odd. 

Now that A is even-times even is manifest ; 
for it has not its half odd. [vu. Def. 8] 

I say next that it is also even-times odd. 

For, if we bisect A, then bisect its half, and do this con- 
tinually, we shall come upon some odd number which will 
measure A according to an even number. 

For, if not, we shall come upon a dyad, 
and A will be among those which are doubled from a dyad ; 
which is contrary to the hypothesis. 



4«> t.: BOOK IX [ix. 34, 35 

Thus A is even-times odd. 

But it was also proved even-times even. 

Therefore A is both even-times even and even-times odd. 

Q. E. D. 



Propositiok 35. 

If as many numbers as we please be in coniinved proportion, 
and there be subtracted from the second and the last numbers 
equal to the first, then, as ths excess of the second is to the 
first, so will the excess of the last be to all those before it. 

Let there be as many numbers as we please in continued 
proportion, A, BC, D, EF, . .• , 
beginning from A as least, ' " a- 
and let there be subtracted ' ' B-^0 
from BC and i^^the numbers ° 



BG, FH, each equal to A ; e -^ j^F 

I say that, as GC is to A, so 
\% EH \.Q A, BC, D. 

For let FK be made equal to BC, and FL equal to D, 

Then, since FK is equal to BC, 
and of these the part FH is equal to the part BG, 
therefore the remainder HK is equal to the remainder GC. 

And since, as EF is to /?, so is /? to BC, and BC to A, 
while D is equal to FL, BC to FK, and A to FH, 
therefore, as EF is to FL, so is LF to FK, and FK to FH. 

Separando, as EL is to LF, so is LK to FK, and KH 
to FH. [vii. II, 13] 

Therefore also, as one of the antecedents is to one of the 
consequents, so are all the antecedents to all the consequents ; 

[vii. I a) 

therefore, as KH is to FH, so are EL, LK, KH to LF, 
FK, HF. 

But KH is equal to CG, FH to A, and LF, FK, HFto 
D, BC, A ; 
therefore, as CG is to yj, so is EH to D, BC, A. 

Therefore, as the excess of the second is to the first, so is 
the excess of the last to all those before it 

:• • Q. E. D. 



IX.3S. j6] PROPOSITIONS 34—36 411 

This proposition is perhaps the most interesting in the arithmetical Books, 
since it gives a method, and a very el^;ant one. of summing any serin of 
terms in geometri^ai progression. 

Let a,, a,, a,,...a^, a,^., be a series of terms in geometrical progression. 
Then Euclid's proposition proves that 

(«ii+i-Oi> : (a, + i*. + ... +a,) = (o,-fl,) : Ui- 

For clearness' sake we will on this occasion use the fractionul notation of 
algebra to represent proportions. 

Euclid's method then comes to this. 

Since ??!!=.?!_= ... = «», ' ' ^■ 

we have, separandi), 

"n*! — °ii _ ^-''n- i _ _ tfj — i»a ^ gj-tfi 

whence, since, as one of the antecedents is to one of the consequents, so is 
the sum of all the antecedents to the sum of ail the consequents, [vti. 12] 

<^%*\ - "1 ^ I)-"! 

which gives a, + «,+ ... 4- a,, or S.. 

If, to compare the result with that arrived at in algebraical text-books, we 
write the series in the form 

a, ar, ar',.,.ar'~' (n terms), 
a>* ~a ar~a 



we have 



5, a 

alr'-l) 



s,= 



Proposition 36. 

jy as many numbers as we please beginning fr(nn an unit 
be set out continuously in double proportion, until the sum of all 
becomes prime, and if the sum multiplied into the last make 
some number, the product will be perfect. 

For let as many tiumbers as we please, A, B, C, D, 
beginning from an unit be set out in double proportion, until 
the sum of all becomes prime, 

let E be equal to the sum, atid let E by multiplying Z? 
make FG ; 
I say that FG is perfect. 

For, however many A, B, C, D are in multitude, let so 
many^, HK^ L, Mh& taken in double proportion beginning 
from £ ; 
therefore, ex aeguali, as A is to D, so is E to M. [vii. 14] 



4« BOOK IX [1x36 

Therefore the product of E, D is equal to the product of 
A, M, [vii- 19] 

And the product of E, D is FG ; 
therefore the product of ^4, M is also FG. , ; r•i^'^ 

Therefore A by multiplying M has made FG ; 
therefore M measures FG according to the units in A. 

And ^ is a dyad ; 
therefore FG is double of M. .^ 



— A 



!= E 

M 

F 1 Q H 



Q- 



But M, L, HK, E are continuously double of each other ; 
therefore E, HK, L, M, FG are continuously proportional in 
double proportion. 

Now let there be subtracted from the second /f/C and the 
last FG the numbers /W, FO, each equal to the first E ; 
therefore, as the excess of the second is to the first, so is the 
excess of the last to all those before it [ix. 35] 

Therefore, as //JC is to E, so is OG to M, L, KH, E. 

And NK is equal to E ; 
therefore OG is also equal to M, L, HK, E. 

But FO is also equal Xa E, . 

and E is equal to A, B, C, D and the unit 

Therefore the whole FG is equal to E, HK, L, M and 
A, B, C, D and the unit ; 
and it is measured by them. 

I say also that FG will not be measured by any other 
number except A, B, C, Z>, E, HK, L, M and the unit 

For, if possible, let some number P measure FG, 
and let P not be the same with any of the numbers A, B, C, 
D, E, HK, L, M, 

And, as many times as P measures FG, so many units let 
there be in i? ; 
therefore Q by multiplying P has made FG. 



IX. 36] PROPOSITION 36 413 

:i But, further, £ has also by multiplying D made FG ; 
therefore, as ^ is to 0, so is /* to D. [vn. 19] 

And, since A, B, C, D are continuously proportional 
beginning from an unit. 

therefore D will not be measured by any other number except 
A, B, C. [«. 13] 

And, by hypothesis, P is not the same with any of the 
numbers A, B, C; 
therefore P will not measure D. 

But, as P is to Z?, so is ^ to i2 ; 
therefore neither does £ measure Q. [vii, Def. ao} 

And £ is prime ; 
and any prime number is prime to any number which it does 
not measure. [vn. >9] 

Therefore E, Q are prime to one another. 

But primes are also least, [vii. *i] 

and the least numbers measure those which have the same 
ratio the same number of times, the antecedent the antecedent 
and the consequent the consequent ; [vn, 30] 

and, as £■ is to 0, so is /* to Z? ; 

therefore £ measures P the same number of times that Q 
measures D. 

But D is not measured by any other number except 
A, B. C; 
therefore Q is the same with one of the numbers A, B, C. 

Let it be the same with B. 

And, however many B, C, D are in multitude, let so many 
£^ HK, L be taken beginning from £. 

Now £, HK, L are in the same ratio with B, C, D\ 
therefore, ex aequali, as ^ is to D, so is ^ to Z. [vn, 14} 

Therefore the product of B, L is equal to the product of 
A E. [vii. 19] 

But the product of D, E is equal to the product of Q, P; 
therefore the product of Q, P is also equal to the product of 
B,L. 

Therefore, as is to B^ so is L to P. [vn. 19] 

And Q is the same with B ; 
therefore L is also the same with P : 



434 BOOK IX [ix. 36 

which is impossible, for by hypothesis P is not the same with 
any of the numbers set out. 

Therefore no number will measure FG except A^ B, C, 
D, E, HK, L, J/ and the unit. 

And FG was proved equal to A, B, C, D, E, HK, L, M 
and the unit ; 
and a perfect number is that which is equal to its own parts ; 

[vii. D«f. 2»] 

therefore FG is perfect, 

Q, E. i>. 

If the sum of any number of terms of the series 
I, i, a*, ... 2""' 
be prime, and the said sum ( e multiplied by the last term, the product will be 
a "perfect" number, i.e. equul to the sum of all its factors. 

Ijet I + I + j' + . . . + a""' (= 5,) be prime ; 
then shall S^ . 2""' be "perfect," 

Take {it - i) terms of the series 

•->ii> ^"Jii, 2 Ob, ... i Ob. 

These are then terr.is proportional to the terms 
a, 3', 2', ... 2"-'. 

Therefore, ex aequali, . , 

jra— = ^;a-^j;, , [vii. 14] 

or 2 . 2"-'5',= j"-' . J,. ■ [vii. 19] 

(This is of COttrse obvious algebraically, but Euclid's notation requires him to 
prove it.) 

Now, by IX. 35, we can sum the series S^ + 2 j^ + ., . + 2"-'i^, 
and (25. - S^) : 5. = (a-' 5. - .S;) : (.S; + a^; + ... + a—3.). 

Therefore .S; + 25. + 2»5, + . . . + 2*-'5, = a"-'5, - S,, 
or s"-"^, = 5, + a5. + 2*5, + . . . + 2*-*Sn + J. 

= S^+tS„+ ... + 2'-»5, + (i + 2 + 2' + ... 4- a"-"), 
and 2""' 5, is measured by every term of the right hand expression. 

It is now necessary to prove that a*"'.?, cannot have any factor except 
those terms. 

Suppose, if possible, that it has a factor x difTerent from all of them, 
and let 2'"''S'» = x . ut. 

Therefore S^^: m = x : 2""'. [vii. 19] 

Now 2"~' can only be measured by the preceding terms of the series 
I, 2, a',.., a"-', [IX. 13J 

and X is different from all of these ; j _ 

therefore x does not measure a""', 
so that S, dots not measure m. , [vii, Def. 20] 

And S, is prime: therefore it is prime to m. [vii. 29] 

It follows [vii. 20, 2r] that 

m measures a""'. 



IX. 36] PROPOSITION 36 4^5 

Suppose that > <■ w = J^ .i- 

Now, ex aequaii, a' r i""' = S, : j"-'-' 5,. ' ' 

Therefore a'' . a"""-' 5« = s""' 5. [vji. 1 9] 

- X . m, from above. 

And m = a'' ; 
therefore .r = 2"-'-' S», one of the terms of the series Sx,^S^,2*Sf, .,.i*~'S^: 
which contradicts the h)pothesis. 

There a"~'5, has no factors except 

5„ zSn, a'5„ ... i"-'^., I, 2, a', ... i'-\ 

Theon of Smyrna and Nicomachus both define a " perfect " number and 
give the law of its formation, Nicomachus gives four perfect numbers and no 
more, namely 6, 28, 496, 8138. He says they are formed in "ordered" 
fashion, there being one among the units {i.e. less than 10), one among the 
tens (less than too), one among the hundreds (less than 1000) and one among 
the thousands {less than loooo) ; he adds that they terminate in 6 or 8 
alternately. They do all terminate in 6 or 8, as can easily be proved by 
means of the formula (2"- 1)1""' {cf. Loria, Le tcienst tmtte neW antica 
Grtda, pp. 840 — i), but not alternately, for the fifth and sixth perfect numbers 
both end in 6, and the seventh and eighth both end in 8. lamblichus adds 
a tentative suggestion that perhaps there may he, in like manner, one perfect 
number among the "first myriads" (less than loooo'), one among the "second 
myriads" (less than loooo'), and so on. This is, as we shall see, incorrect. 

It is natural that the subject of perfect numbers should, ever since Enclid's 
time, have had a fascination for mathematicians. Fermat (160 1—1655), in a 
letter to Mersenne {CEuvres de Fermai, ed. Tannery and Henry, Vol. 11., 
1894, pp. 197 — 9), enunciated three propositions which much facilitate the 
investigation whether a given number of the form a"-! is prime or not. If 
we write in one line the exponents i, a, 3, 4, etc. of the successive powers of 
2 and underneath them respectively the numbers representing the correspond- 
ing powers of a diminished by i, thus, 

1334567 S 9 10 II ...n 
I 3 7 IS 3' 63 "7 ass S" >«23 2047...2"-i, 
the following relations are found to subsist between the numbers in the first 
line and those directly below them in the second line. 

I. If the exponent is not a prime number, the corresponding number is 
not a prime number either (since a" ~ i is always divisible by a' — i as well 
as by a' - 1 ). 

a. If the exponent is a prime number, the corresponding number dimi- 
nished by I is divisible by twice the exponent, [(a*- a)/2M = (a""' -i)/« ; so 
that this is a special case of " Fermat's theorem that, if/ is a prime number 
and a is prime to/, then cf~' is divisible by/.] 

3. If the exponent n is a prime number, the corresponding number is 
only divisible by numbers of the form {tmn+ i). If therefore the corre- 
sponding number in the second line has no factors of this form, it has no 
inte>;ral factor. 

The first and third of these propositions are those which are specially 
useful for the purpose in question. As usual, Fermat does not give his proofs 
but merely adds : " Voilil trois fort belles propositions que j'ay trouvees et 
prouv^es non sans peine. Je les puis apfteller les fondements de I'invention 
des nombres parfaits." 



496 ^ BOOK IX [ix. 36 

I append a few details o( discoveries of further perfect numbers after the 
first four. The next are as follows : 

fifth, 3"(3''-i) = 33 SSO 336 •' f 

sixth, »" (*■'-!) = 8 589 869 oj6 

seventh, i"(i'*-i)= 137 438 691 jaS 

eighth, 2* (2"-!)= 2 305 843 008 139 95* ia8 

ninth, 2" (2*' - r ) = 2 658 455 99 1 569 83 1 744 654 691 6 1 5 95 3 842 1 76 

tenth, a"(a"-i). 
It has further been proved that s'"- 1 is prime, and so is a'"- r. Hence 
3'" (a'"-!) and 3'"(2"'-r) are two more perfect numbers. 

The fifth perfect number may have been known to kmblichus, though he 
does not give it ; it was however known, with all its factors, in the fifteenth 
century, as appears from a tract written in German which was discovered by 
Curtze (Cod. lat. Monac. 14908). The first eight perfect numbers were 
calculated by Jean Frestet(d. 1670). Fermat had stated, and Euler proved, 
that a"- I is prime. The ninth perfect number was found by P. Seelhoff 
{Zatithrift filr Math, u, Physik, xxxi., 1886, pp. 174 — 8) and verified by 
E, Lucas (Mat/Usis, vii,, 1887, pp. 45 — 6). The tenth was discovered by 
R. E. Powers (see Bulletin of ifu Ameriam Mathematical Society, xvni,, 191*, 
p. i6a), 2"*— I was proved to be prime by E. Fauqueuibergue and R. E. 
Powers (191 4), while Fauquembergue proved that 2""-! is prime. 

There have been attempts, so far unsuccessful, to solve the question 
whether there exist other " perfect numbers " than those of Euclid, and, in 
particular, p>erfect numbers which are odd. (Cf. several notes by Sylvester in 
Comptes rtndus, cvi., t888 ; Catalan, " Mdlanges mathrfmatiques " In M^m. de 
la Sec. dt Liige, 2* S^rie, xv,, 1888, pp. ao5 — 7 ; C. Servais in Mathisii, vii., 
pp. 228 — 30 and VIII., pp. 92 — 93, 135; E, Cesiro in Math^sis, vii., 
pp. 245 — 6 ; E. Lucas in Mathisis, X., pp. 74 — 6). 

For the detailed history of the whole subject see L. E. Dickson, History 
o/the Theory of Numbers, Vol. 1., 19 19, pp. iii — iv, 3 — 33. 






INDEX OF GREEK WORDS AND FORMS. 



dxpot, extreme {of numbers in a series) 3^3| 
367: Axpor xol fUvof \byv rtTfi^ffOu, *'to 
be cut in citrcme Ewd mean ratio*' 1S9 

aAo-yoj, irrational 117-8 

iraXoyiSj proportiiM); definition* oft inter- 
polated 1^9 

dviXjryovsdfd \&yov, propoTtLonal or in pro- 
portion: used as Lad^clmable adj^ and as 
adv. r^pt 165: pjiffn drdkoyof^ mean pro- 
portional (of straight line) 119, similarly 
^ff« AwA^aryor of numbers 195^ ^6^ etc. : 
rpfrq (rpJrot) 6*6Xey», third proportional 
4I41 4O7-S: TtrAfT^ [riritfiTtH) AtdXoyoi/j 
fourth proportional 115,409: ^f 1^ drdAo-^op 
in continued propottion 346 

4j'd'rd?\^»{}<,6yot)j inverse (ratio) ^ inversely t34 

dya^Tpwftij XA7W, conversion of a ratio 135 
dFi^dxit drurdjTLt Cirdt, unequal by unequal 

by equal (of solid numbers) = scaltrtUt 

t^fjwlvKojf ff^xiffKot or pvtd^Ktn 190 
dra^iifr TtrayfUrvr TUtr ^^yvf (of perturbed 

prGp^tiQn) in Archimedes 136 
d»Ta>aJ!pca'LTt i4 <l^7H>|i> definition of SQftti raiia 

in Aristotle (d^tfu^o^^ii Alexander) no: 

tcrmi expLained m 
dmirEirorf^^TtL v^^^i^juara, reciprocal (^reci* 

procaily related) figures, interpolated def. 

off 189 
dirXamSt, brcadthlcss (of prime numbers) 185 
dTOJcarHTartjcAt} neumni {=jpheri£al}, of 

numbers 391 
drrftftfoff to rrneti occasionally to tottiA 

(instead of i^vrtffffm) ^i a]so=to/arf 

ikrpHgh, to ii> tfA 70 
dpiff^, number^ definitions of^ 380 
i^i&Kii Apviot^^.flOl^ (Nicouiachus) 183 
dpr^dnr i/)T4<Hj £vtn-timts even i8i-a 
d/mdivtr irf/R(ro-d(^ even-times odii ^83-4 
4/moF^«TTot> tfFm-tfdtf(Niconiacbusetc.) iS* 
4>)Ti(>i (d^^^i)^ ev«n (number) 181 
4«^£p0rmj (prime and) locomposite (of 

numbers) 184 

^f^jr^nt, to stand {of anfle standing^ on 

circumrercnce} 4 
^fiiffxor, ^itiir-skaped (of "scalene" solid 

niunben} 990 

7ryor^B(in constnictionB), **l«t it be made" 
^48 



Tfywif Jbf rfi) Ti iirtTaxW"! "what was en- 
joined will have been dune" Bo, itit 

7c»A^»Dtt 4 ^£ n^ur, *' their proditet** 316, 
336 etc,: h Ik toD ^pit y(?fi;iej'Of='* the 
tptare of the one " 317 

ywi&fivrt gnomon: Democritus irf/4 Atatpo- 
p^ 7ifc3Jiai'« (T^tiitiiji or yufliit}) ij npt 
^atStf-jDt jti^jcXov Kol f^oipift 40: (of numbers) 
189 

-/pafifitK6f, linear (of numb^ in one diinMn-> 
ston) 1S7: (of prime numbers) 185 

ypdiptv ffatt ^'lo be proved" (Aristotle) tta 

St&rtpot, see&nd&ry (of numbers): in Nico- 

machus and lamblichus a subdivision of 

odd 186, 387 
j^fl^^^ycw, **admittLnf ^' (of segment of circle 

admitting or contaming an angle) 5 
Jnu/Kr^rtfoj (used of ^^ separation" of ratios): 

jfoipf^^n-a, seporand^y opp* to t\^KttfiX¥v,t 

£&mpcn£nd& 16B 
itatptffit \^you, separfttiott, literally division^ 

of ratio 135 
^inimrffUrTi {ifia\oyla), disjoinedi = discrete 

(proportion) 2^^ 
ii^HwTi, separanda^ literally dividends {cH 

proportions) 135 
ii^py^iUt^ (draXfi^^a), discrete (proportion)^ i.e, 

in four termj, as distinct from continuous 

{avwt.x¥i <nmj/i^Fif) in MrM terms [3i;| 

393 
ii-ffxB^ (AidytfLp), "let it be drawn through** 

or *' aerass** 7 
&^ {!(r0Uf«: d/i^va/t (of ratios) 156: i^i' trov ^ 

rrrapayfi^nff draAc^^^, ^Vj<^ aeffuati in per- 

lurbed proportion" 136 
JtffiXoupof, twice-iruncaied (of pyramidal 

numbers) 391 
A^TAdfTioT \6y<ftt double ratio : ffiK-Xatfibv 

X^^ott dupiicaie ratio, contrasted with, 133 
Sf^a^r^ posver: =actual value of a aub- 

multiple in units (Nicomachus) i8t ; ^side 

of number not a complete square (i.e. roctt 

or surd) in Plato ^88^ ^90: =square in 

Plato 194-5 

fWw, figure 134: =form 254 
fvotfroff each: curious use of^ 79 
fWiiptfiA, defect (in application of areas) 161 
iXkeiww^ *"fall shtirt" (in application of 
area$) i((i 



4^ 



INDEX OF GREEK WORDS AND FORMS 



tftMi-rr^tM^ /all in ( = bc interpolated) 35B 
ba rXe^, ''several fftut" (def, of tiiunber) 

/roXMf (\4yM),altenute {ratio): aitemntely* 

&iternand& [14 
^op/i^if^ ^^pi >'» (active) I IV, Def. 7 «nd 

IV. I, 7<h 80, &i 
jrrb, within (of internal contact of cinles) 

13 . . 

^^ ^dXo7or} in continued proportion (of 
tcnn^ in geometrical progression) 546 

^ri^jHor X^TQii suptfpartuularU ration 
= the ratio Jb+ [): Bv 195 

^irfiTffAot (4^0^}, plane (number) 387-S 

^6^ra, conse^^uenla ( = "foUowing" terms) 
in a proportion 134, ijS 

ire/Mjiii)jriff| obiong (of numbers): in Plato 
= r^>4^T^, whicb however is distinguished 
from /rt^^^Kijf by Nicomachus etc. 189- 

t06vypafituii&tf rectilinear (term for prime 

numbers) ^85 
*d*y/itT^ifdf, eulhymetric (of primes) ^85 

^^fX^^faT antecedents ("leading" terms) in 

a proportion (34 
TWtpt than : construction after 6tr\aait«f etc. 

tSw/iiiir^t of jsquare number (lamblichus) 193 
iff^tt tffdxit Iff Off equal multiplied by equal 

and again by equal (of a cube number) 

490, ^gt 
Itrim tttojt equal multiplied by equal (of 

a square number) 391 
fo-cUif tffoj 4kKTT0*ij(tt (^ifbpint), species of 

solid numberst = T\ip0lt [Svitt or mfMT) 

KoKttffBvj "let it be calledt" indicating origi- 
nality of a definiUon 119 

KaTOfLerptuf^taeAttite uj: without remainder, 
completely {irXifpovrTaH] 380 

itaTauKtvd^Ut construct: rCtr aOrvf ttaTa- 
7«CMi(r#^Twv^ *' with the ^me construc- 
tion *' 1 1 

KararotA^ irap^^t, Secfh cantmis of Euclid 

ttirrpowi centre: tj Ik rw K. = radiui 2 
KtptxTOtiSTi^ ywpiokt hortttiki angle 4, 39, 40 
JcXaCt to *rff*t^j iij^fiT/! ffCKXi^rfffj ffi^ irdXcf 

47 ; HjrXdd-tfcu^ def- of^ alluded to by 

AriF^olIf: 47 
xMoupof, trufKoied (of pyramidal number 

mitmi vertex) igi 
Kvg\u6t, cyciici a particular species of square 

number 191 

Xd^ot^ ratio: meaning 117 : definition of, 
11&-9: original meaning (of someLbing 
€xpressed) accounts for use of AXoT^ri 
having no ratio, irraiiatial 117 

fu^ov^6a.i^ to be isoIiUidf of ftori^i (Theon 
of Smyrna) ^79 



fiipofj part; two meanings 115: gencrallys 

submultipte 380: m'^, parfj (^proper 

fr«ction) 115, 180 
fii^ AriXtyot (td^cla), fJrof driXoyof {Api9* 

fiM), mean proportional (straight line or 

number) 119, 195, 363 etc. 
fi.ii ydffy "suppose it i> not" 7 
fi^Kot, length (of ntimber in one dimension) : 

= side of complete square in Plato 188 
tu»Att unit, moniad: supposed etymological 

connexion with fi6m, solitary] ftot^t rest 

379 

5^i«; similar: (of lectilinod figures) 188: 
(of plane and solid numbers) ^93 

^fiiotAnfs Xiytaft '^flimilarit^ of ratios" (mier- 
polated def. of proportion) 119 

ifiiikayoi^ homologous, corresponding 134; 
exceptionally "in the same ratio with" 

Spat, Urm, m a proportion 131 

rapo^dXXet^ ^70, used, exceptionally, instead 

of wapa^AWfiy rapk or draypd^civ A.t6 161 
irupaXXdirru, *'faJl sideways" or '*awry" 54 
mvTAypafiitAoy 99 
wtpalr9wa twAttp, "limiting quantity" 

(Thymaridas* definilion of unit) 179 
vepi^t/d^Kij ApriQi, edd'iirttii tvcn 181-4 
^tpuj^ibcit irtpiffff6i, odd-timts odd 184 
r«^(r4pT-mi^ otkleven (Nicomachus etc-) 3B3 
■rtputv^, odd tSi 
irifMjraft how grei^t: refers to centifttdom 

(geoTneErical) magnitude as woff&i loditftvte 

(multitude) 116-7 
T^SiKirift, used in v. Def. 3, and spurious 

De^^ 5 of V] . : = n^ (not quanhipiiiiiy as it 

is trarulated by Pe Morgan) 116-7^ 1S9- 

00: supposed mukip1icationofi-)t\ur^iTi7Tct 

\vu Def. 5) r3i; dislinclion between 

T7Auifin7f and fHytB^i 117 
rXdrfrt, breadth: (of numbers) 1S8 
irXeirpd, side: (of ^tors of "plane'' and 

'* solid" numbers) 188 
vXi^ti^or uptfffiiiitov or rtr^paa^ixdur^ defined or 

finite multitude (delinition of number) tSo: 

4k ^tofdlup ffvytelfunv irK^Btn (Euclid's 

def.) 38a 
r^XXarXuridj'f^r, multiply : defined tBj 
xoXXavXaffiOff^df I multiplication : itad' 6wotof- 

Qijr v4XXarXwr<a^i4^ "(arising) from any 

mukiplJcjLtion whatever*' no 
iroXXairXd<riaft multiple ; i^Ajcit roXXurXdirca, 

equimultiples 110 etc. 
Trok6r\tupor, multilAtcrah excludes Ttrpir 

rXfupoi^, quadrilateral 339 
iropitrciffdatt \o ^ttd 148 
irod'tijciT tfurAKii -roo^i "so many times so 

many limes so many*' (of solid nambers, 

in Aristotle) i86t 1190 
•r9oi.Ki.% TTOffot^ ^'so many times so rruny^' (of 

pbne numbers, in AHstotle) 186 
TiKT^i qvantily, in Aristotle iij: refers 

to nmltimde as t^Mkoi' to magnitude 

1 16-7 



INDEX OF GREEK WORDS AND FORMS 



429 



TTpQfiiitnti, obhnj; (of numbers): in Plato = 

^*^^ifijj» but disiinguished by Nico- 

iTuchus etc. ^89-901 iQ.i 
TpHra'a7^^«, tOdiiraicrm At: (of a circle) to 

ismpUti^ when segment is given 56 
vpcvfifptii'i to £nd in addition (of finding 

tbird and fourth proportionals} 114 
r/Ktffor, prime 1*4-5 
TpuTDi r^ dWi^Xot^T, (numbers) prime to 

one anotber 4B5-6 

^ifr^, rational (literally '* expressible") T17 

"J'rcnir continuous : ^vj^x^ d*ttXoyft| '* oon- 
tinaoui proportion" (in three terms) 131 

runf^jL^i^ duHiXryia, ccnmcted {i-e. con- 
tlnuotis) proportion 131^ «j : <fwijft^rof, 
of iompsmHa ratio in Archimedes 1^3 

ffVitBirrt, compOMnda 1 34-5 

rirr0f<rit XE^yair, composition of a ratio, dis- 
tinct from iampoundtHg ot ratios 134-5 

fft^fffTflT, composite (of numbers) : in Nico- 
machus and lambUchus a subdivision of 
odd 366 

tfittrtfif/Uj n^xei^oi {of ratios) 135, iSg-90: 
ffvyifi^^PR and ^taip^0iw7Vk {c^mpontndo and 
ji^mH^) u^ed relatively to i^ne another 

fftfffTij;ia fi/^SuTr "collection of units" (def. 

of number) a So 
ffvimffiaTtK6tf collective 379 
ff^<u^K6u spMerjca/ (of A particular spiecies of 

cube dumber) 391 
f^it^LvKot, or ff^rjtrlffMoi, of solid number 

with all three sides unequal (= '* scalene") 

7tp 



ffX^ffu* ** relation '* ; wwA rxi^rni '* a sort of 
relation" {in def* of ratio) ji5-7 

TaA-np4icifl^ of square number (Nicom,) ^95 
TaCr&nji \&yuWi "sameness of ratios" 119 
riXfioti pfr/ec/ (of a class of numbers) 193-4 
TnTat^ivyj {iiiahoyla), *' ordered (proportion) 

Ttrttpayttinf AraXoyiAf perturbed proportion 

136 
TfrpdrXtu/Kii', qu4dril*teral» not a " polygon" 

Tfi^jCia (Ai>H\«;)t segment {of circle): r/i^jmirn 
7C4*'fa, angle of ^ segment 4; I9 r^^juan 
7urr^ angle in a segment 4 

Tp^e^ (ffi[!jf\«'}f sector (of circle): (TKurvrft- 
^wftr Tfl^^, "shoemaker's knife" 3 

r^^At^l-^i i^i figure}^ mtor-likt 5 

rG<ravrarXdiriov, "the same multiple** 146 

TpLyijror : r^ rptrKw*, rh ^* dXX^Xur, ttiplei 
inttrwoven triangle, = pentagram 99 

TpLv\a.trt^^ triple, TpiTXnffW, triplicate {of 
ratios) 133 

riffX^^^^t happen: 4XXa, cC frio(<c» £(ruia 
To^Xaa-Xd^fai "other, chance, equimulti- 
ples" 143-4^ ryxoLtfii 7i*if(«» "(jwi'ttigle'* 



uirr^<Xif$ or virrprAtuff, "over-perfect" (of 

a class of numbers} 193-4 
(iro3(TX«rtor, st^dupiicate^ = half (Nico* 

machus) «8o 
AraT«X\iiYX^u)r, submulttple (Nioomachus) 

iSo 
0^r» height [89 



Xu^fVt av^a. 354 



ENGLISH INDEX. 



Adnutai 199 

Alcinotu 96 

AiUms/e and a/temateiy (of ratiiK) 134 

AUeituttire prtxifa, interpoUtcd [a. uu 9 
utd following) 11 : thmt in ui. to cUiuked 
by tttron 93-4 

Amaldii LJgt>, 39* 11$ 

Amdipi^tii eate of VI. y, joS-y 

ADuiimander 1 1 1 

Anaximenu iii 

Angle : in|r!es not leu thin two right an^^les 
not recogni$«d as tingles (cf. Heron, 
ProcluSf Zenodorus) 47-9: haUcw^angied 
figure (the le-cnLiant uigle was exf^n^ 
48; dia Eudid eictend ^^ angle" to angles 
gremter than two right angles in VI. 33? 
*7J-6: '* angle a/ semicircle" and " t^f 
legment" 4: herniiif ttig\t 4, ^9, 40 : 
controversies a.bont *' an^le of semicircle ** 
and ^bm/i^f angle 39-41 (see alsto JfernUki) 

AtUue^nti (le^ing terms in proportion) i J4 

AiidparalLels : may be used for construction 
of VI, II, 115 

Apolloaius : ^/iaixf firvmri problem fromi&T, 
lemma by Pappus on, O4-5 : FUnt Loci, 
theorem from (arising out of EacL vt, 5), 
also found in Aristotle 198-900: 75, 190, 

'*? . 

Applicatii>n of areas (including txcitdmg and 
jaiimg short) corresponding to soLutioo of 
quadratic equations 167, 358-6o» ^63-5, 
166-7 

AppToaimations t 7/5asappTOKimattDn Xo^J^ 
(Pythagoreans and Plato) 119: appruxi- 
mations to J^ in Archimedes and {in 
sexagesimal fractions) in Ptolemy 119: to 
T (Archimedes) j 19 : to V4300 (Theon of 
AleEandria) 119 

Archimedes: MetAod <j{, 40: Li&tr aifump- 
ieruiHi proposition from, 6^ : approxima- 
tions to 1^3, square roois of large numbers, 
and to V 1 19: extension of a proportion in 
aymmensurablesto cover inoommcnsur^bles 

Archytas : proof that there is no numerical 
geometric mean between n aiid m Hh i 495 

AnMotle; indicates proof (pre- Euclidean) that 
angle in semicircle is ri^bt 63 : on def. of 
same ratia {=sami drrara^wtf) I90*i : 
<m proportion as ' ' equality of ratios " 1 19 : 



on theorem in pfoportion not proved 
pfMfra//y till bis time iij: on proportion 
in three terms (irurfx^^i continuous)^ and 
in four terms (0ivpfM^« discrete) I3r»l93 : 
OD aitimeUt ratios 134: on jflwr^c ratio 
t34i 149- on similar rectilineal figures 18S: 
has locus- theorem (arising out of Euct- vi. 
5) also given in Apollonios' Phn^ Loci 
198-900: on unit ^70: on number 180: 
on non-applJcability of arithmetical proofs^ 
to magnitudes if these are not numbers 
1 1 j : on deAnitions of odd and even by one 
another i%i'. on prime numbers 384-5: 
on composite numbers as plane and solid 
986, 18B, 990: on representation of 
numbers Ly pebbles forming figures 186 
Arithmetic, Ettnunts of, anterior to Euclid 

195 

Auguit, E. F, 33, 3£r 149, 35S, 456, 419 
Austin, W. 174, tSSt in, 959 
AxioBU tacitly assumed * in Book v« 137 ^ 
in Book vii. 994 

Babylonians iii .,_ ^ , 

Baermann^ G, F« 913 

Baltur, R. 30 

Barrow: on EucL v. Def^ 3, 117: on v* 

Def. 5. "i: i^t i36, 938 
Billin^tey, H< 56, 138 
Boetbms 195 
Borelli, G. A. 9, 84 
Breadth (of numbers) = second dimension or 

factor 48^ 
Briggs, H. 14^ 

Camerer, J. G, 99, 95, 9S, J3» 34, 40, 67, 

jui, ijJ» 189, 913, «44 
Campanus 98, 41^56, 9a, n^t '19. isi^ 1461 

)8q, an, 934, 933,353^ *7S, 5w, l^if 318 
CandaLla 189 
Cantor, Monti 5*. 40, 97 
Cardano, Hieronimo 4I 
Case\ Greeks did not in/tr limitii^r caseSf 

but proved them separately 75 
Casev, J. 197 
Catalan 496 
Cesiro, E. 416 
" Ckanct equimultiples^' in phrase ** other, 

chance, et^uimattiples *' T43-4 
Circle: deJtnition of equal ctrclcs 9: circLeii 



43* 



ENGLISH INDEX 



tmichiH^, meaning of definition, 3 : *^ circle*' 
in s^nse of " drcnrnfcrence ** ij; circles 
tntcTsecMng and tonching, difficoltiej in 
Euclid's treaiment of^ ^5-71 18-9, modem 
treatment of, 30-^ 

aavius *, ^i, 41, 47, 49. 53, ^^6, 67, 70, 75, 
130. i79t tgo, 331, 138, 344, 171 

Commandinus 47, 130^ 190 

Ccmp^n^tidQ [avrffh^iy^ denoting '^composi- 
tion " of Ffttios f*v. '. ccmp^tunda and 
stparando nsed relatively to each other 
168, 170 

Composite nnmbei^t in Kuclid i36: with 
EucL and Theon of Smyrna may be even, 
but with Nicom. and Iambic are a sub- 
division of odd t86: plane and solid 
numbers species of* lS6 

** Composite to one another*'^ (of numbers) 
586~7 

Composition of ratios (ffiSvflrtftt XAyoir), de- 
noted by ttunpimfndo [fftn^^i^ri), distinct 
from compounding ratios 134-5 

Compound ratio: explanation of, 133-3: 
questionable definition of^ T&9-io: com- 
pounded ratios in v. 90-'>3, 17&-8 

Constqitmti {"following" termi in propor- 
tion) 134, 138 

Ccntinucus proportion [ffv*rxift or ^uvi^^f^v^ 
diuXpyia) in three terms 131 

Conversion of ratio (d»urrfH»^7r Xd^ou), de- 
noted by f^nvert^ndo f^uuTTpi^ffatTi) 135 : 
ionvertittdo theoreni not eat^blished by v. 
19, Por. 174-51 hut proved by Simson's 
Prop. E 175 

C&twcrtgndi> denoting " conversion " of rati<}5, 

€orr€ipcnding magnitudes 134 

Cube: duplication o^ reduced bj Hippo- 
crates to problem of two mean pro- 
portionals 133 : cube number, def, of, 191 1 : 
I wo mean proportionals between two cube 
numbers, 194^ 364-f 

Curtze, M. 446 

CysUii ofa particular kind of aqtiare number 

CymtmUhia of teotaud 44 

Deda of Euclid \ Def. 1, 348 : Prop. 8, 349- 
50: Prop. 44, 346-7: Prop. 55, 354: 
Props, 50 and 68, 349: Prop, f8| 363, 36 j : 
Props. 59 and 84, «C6-7 : Prop. 67 assumes 
part of converse of Simson^s Prop. B (Book 
vl) 3«4 : Prop. 70^ 750: Prop* 85, 364 : 
Prop. 87, 338: Pirop, 93, %%i 

Dechales 359 

Dedekind's theory of irrational numbers 
ctirresponds exactly to Euch v. Def, 5, 
134-6 

Democ ritu & : " On diferermt e/gnfff/ien * * etc. 
(? on *^ angle of contact ") 40 : on parallel 
and infinitely near sections of cone 40: 
slatedj without proving, propositions about 
volumes of cone and pyramid 40 

De Morgan, A. : on definition of ratio 1 16-7 : 
on extension of meaning of r-titii> to cover 



incommensurables 118: means of expres<i- 
ing ratios between incommensurables by 
approximation to any extent 118-9: de- 
fence and explanation of V. DeC 5* '^^-4: 
on necessity of proof that teste for greater 
and less, or greater and equal, ratios can- 
not coexist 130-1, 157 : on compound ratio 
[33-3, 334: sketch of proof of existence of 
fourth proportional (assumed in v. 18) 17 1 x 
proposed (emma about duplicate ratios a^ 
alternative means of proving vi» 33, 146-7 l 
5( 7, 9-10, II, 13, 30, 33j 39, s6, 76-71 
83, 101, 104, U6-9, 130, 130* i39» *+5» 
197, 103, 117-81 '3«* '33* '34p »?'*■ *75 

Dei'cyllides li i 

Dickson, L^ E. 436 

Dhrismus for solution of ft qumOratic 159 

DistnU proportion, di^pTifU»it ot Sitj^tvyft^rif 
draX^^Ja, in four terms 131, 393 

*^ Dissimilarly ^rdir%d ^' proportion idro^lur 
rrr^'^pAwiay rmv Aiywf) in Archimedcii 
=r * * perturbed proport ion " 1 36 

Dividittdfi (of ratios}, ae Separation, upar- 
ando 

*' Division (of ratios)/^ jw Separation 

Divisumi {qfjSpurij]j On, treatise by Euclid, 
proposition from, 5 

Dodecahedron : decomposition of faces into 
elementary triangles, 98 

Dodgson, C L. 48, 375 

Duplicate ntio 133: iliirXturJtfv, duplicate^ 
distinct from ttrXwws, double ( = ratio 
1:1), though use of terms not uni Form [33: 
"duplicate^" of given ratio found by VJ- 
11, 3E4: lemma on duplic;tte ratio as aU 
tertiative to method of vi. 33 (De Morgan 
and others) 143-7 

Duplication of cube : reduction of, by Hippo- 
crates, to problem of finding two mean 
pToptortionals [33 : wrongly supposed to 
be alluded to in Timaeus 33 A, B, 394-5 n. 

Egyptians 1 1 3 : Egyptian view of numher 180 

Ennques (FO and Amftldi (U,) 30, 116 

Equimultiples: '*any equimultiples what- 
ever/' iffcUtr raXXttTXafft* Kotf hiri>iQiK>w 
Trdk\aT\a<na(Tfiioy no: stereotyped phrafie 
"other, chancet equimultiples" 143-4: 
should include eiKt each magnitude I45 

Eratosthenes : measurement of obliquity of 
ecliptic (13° 51' 3o")- 111 

Escribed circles of triangle 65, 86-7 

Kudemus 99, in 

Eudoxus 99, 180, 195: discovered general 
theory of proportionals covering incom- 
mensurables [13-3 : was first to jirove 
!»cicntificaUy the propositions about volumes 
of cone and pyramid 40 

Euler 436 

Eutocius: on '* vi. Def. 5" and meaning of 
wij\iK&rqr iiflt 13I1 189-90; gives locus- 
theorem from Apollonius' Plsru Loci 198- 
^00 

Even (number) X definitions by Pythagoreans 
and in Nicomachus ^St : definitions of odd 



ENGLISH INDEX 



*$i 



and «ven by one another unscientific 
{Aristotle) i8t : Nicoin. divide* eien into 
three e1aB$es{l)£2tfH*/fM/j«ivn and (i)^*^- 
tinii't a/d as extremes, and (j) tid-iima 
evtn as intemiediflte ^St-3 

Even-times even* Euclid's use differs fit>m 
use \ij Nicomachun, Theon of Smyrna and 
lamblicbus j8i-a 

Evtn-times odd in Euclid different from even- 
wA/of Nicdmacbus and the lest iSi-^ 

Ex aeptalit of ratios^ t j6 : ex ae^imU pro* 
positions (v. 30, 11), and ex ttegvaU ^^in 
perturbed proportion" (v. 11, 53) t76-8 

Faifofer 116 

Ftuquemberguei E. 41^ 

Fermat, 4Q5, 416 

Fourth proportional : assumption of enistence 
of, in V. iB, and altenuitive methods for 
avoiding {Saocberi, De Morgan, Simsoti. 
Smith and Bryant) 170-4: Clavius made 
the assumption an aiiotn ijo: sketch of 
proof of assumption by De Marfan fji : 
condition for existence of number which 
is fourth proportional to three numbers 
409-1 1 

Galileo Galilei; 00 iiiigit «/ emtaet 41 

Geometric means ^^jsqq*; one mean between 
sqiuire numben 194, 36J, or between 
wmilar plane numbers 3;i-i : two means 
between cube numbers 194, 3^4-5, or 
between similar solid r umbers ^73-A 

Geometrical progression 346 sqq.: summation 
ofn terms of (IX. 35) 410-I 

GheranJ of Cremona 47 

GntHnon (of numbers) 389 

Golden section (flection in extreme and mean 
ratio), discovered by Pytbagoreani og: 
theory carried further by Plato and £u- 
doxua 99 

Greater ratio: Euclid's criterion not the only 
one 130; arguments from greater to le*s 
ratios etc. unsafe unless they go back to 
original delinitioas (Simson on v. lo) i tfi-1 1 
test for, cannot coexist with test for equal 
or less ratio 130-1 

Greatest common measare : Euclid's method 
of finding corresponds exactly to ours 1181 
^99: Nicomachusgivesthesamemethad3oa 

Gregory, D. 116, I43 

Hiibler, Th. loiM. 

Hankel, H. 116, ti; 

Hauber, C. F. 144 

Heiberp, J. L. fiittim 

Henrici and Treutlein 30 

Heron of Alexaivdria: Eucl. It). 11 interpo- 
lated from, 38: extetidslll.io, It toangles 
in segments less than semicircles 47-8: does 
not Tecogr\i5e angles equal to or «eater than 
two rightangles 47-8; proof oflforroula for 
uea of triangle, ti ='Ji (i - o) (r - i) {1 - 1) 
Bj-8: s. l6-t7. »4t >8, 34, 36, 44, iifi, 
189, 301, 310, 383. 39S 



liippasus 97 

Hippocrates of Chios i^^ 

}iorniike angle (vriMTtKtJ^ yvrla) 4, 39, 40: 
^ffm/jilf angle and angle ^ semicircle, con- 
troversies on, 39-41: Proclus on, 39-40: 
DemtKritus may have written on hornlike 
angle 40: Campanus ("not angles in same 
icnse") 41: Cardano (^win/iVi'fj of different 
orders or kinds): Peletier (hertilikt angle 
no angle, no qoanltty, nothing ; angles of 
d^ semicircles right angles and equal) 41 : 
Clawius 41: Vieta and Galileo ("angle of 
contact no angle ") 41 : Wallis (angle of 
contact not incHnaiion at all but degree eif 
curBoiure) 43 

Hultsch, F. 133, 190 

lamblicbus 97, 1x6, 179, aSq, s8i, 3ti3, 384, 
iSj, 186, 187, iSS, 189, 19a, 391. 393, 193, 

4i9> 41.S1 4»* 

Icosahedron 98 

Incommensu rabies: method of testing incom- 
mensurability (process of finding G.C.M.) 
118: means of expression consist in power 
of approximation without limit (De Morgan) 
tig: a]^roximattonK to tji (by means of 
side- and iAg^tntii/-num^n) 119, to ^3 
and to T, 119: to V4300 by means of 
sexagesimal fractions 1 19 

Incomposite (of number) = prime 184 

Ingram!, G. 30, 116 

Inverse (ratio), inversely (drdnAtr) 134: in- 
version is subject of v. 4, For. (Theon) 
144, and of V- 7, Por. 149, but is not 

ETly put in either place 149: Simson's 
B on, directly deducihle from v. 
S, '44 
Isosceles triangle of )V. 10: construction 01, 
by Pythagoreans 97-9 

Jacobi, C. F, A. 188 

Lachlan, R. 11G, 137, m-6, 147, !£(!, 171 

Lardner, D. 38, 35 j, 171 

Least common multiple 336-41 

Legend re 30 : proves VI. t arid similar pro- 
positions in two parts (1) for comment 
su rabies, (1) for incommensurables 193-4 

Lemma assumed in vt, 1%, 141-3: alternative 
propositions on duplicate ratios and ratios 
of which they are duplicate (De Morgan 
and others) 143-7 

Length, ^%« (of numbers in one dimension) 
587: Plato restricts term to side of inte- 
gral square number 167 

Leotaud, Vincent 43 

Linear (of numbers) =( I ) in one diraenston 
187, (i) prime 185 

Logical inferences, not made by Euclid aa, 19 

Loria, G. 435 

Lucas, £. 436 

Lucian 99 

Means : three kinds, arithmetic, geometric 
and harmonic 193-3: geometric mean is 



434 



ENGLISH INDEX 



'* pio^tAoD par exieUetue'' (ki^^im) 191-3: 
one geojuetnc mem between two square 
numbcrSj two between two cube pumbers 
(Plato) 194, 363-5: one geometric mean 
between simiUr plane numbers^ two be- 
tween similRT solid numbers 57'~5- po 
Dumerical geometric mean between n and 
ff + 1 {Archytas and Euclid) 195 

ModerfttuSi a Pytbaeorcan iSa 

MaltipUcation, definition of aS; 

»n-NairtzI 5, 16, j8, 34, 36, 44, +7, 30^, 310, 
383 

Nafliaddln at-TusI ig 

Nestelmuin, G. H> F. 1S7, 193 

Nicomachus iiti, 119, 131, 179, 180, tSi, 
1811, 183, 384, 185, 186, 187, 388, 189, 190, 
»9i, 191, «93, J94, 300, 363, 4»5 

Nlion, K. C. j. 16 

Number: debned by Thales, Eudoxus, 
Modei^tus, Aristotle, EncUd 180: Nico- 
macbuB and lamblicbus on^ 180: repre- 
sented by lidcs 1S8, and by points or dots 
988-9 

Oblong (of number): in Plato citbet 7^^1)^171 
or ^ftp.-fi:nit 388 : but these terms denote 
two distinct divisions of plane numbers in 
Nicomachus, Theon of Smyrru and lam- 
blichos 189-90 

Octahedron 98 

Odd (number) : dcfe- of in Nicomachus 18 1 : 
Pjrtbagoreaii definition iSi : def. of odd 
and even by one another onscientific 
(Aristotle) 181 : Nioom. and Iambi, dis- 
tinguish three cbuses of odd numbers 
(i) prime and inoomposite, (a) secondary 
and composite, as eitremes, (3) secondary 
and composite in itself but prime and in- 
composite to one another, which ts inter- 
mediate 387 

Odd-iiiMi nvM (number) : definition in Eucl. 
spurious 183-4, and dilTers from definitions 
by Nicomachus etc- itid^ 

Odd-iirrus odd (number): defined iti Enct. but 
not in Nioom- and lambL 1S4: Theon of 
Smyrna applies term to prime numbers 
184 

Oenopides of Chios 1 1 1 

" Ordered" proportion {rtrvfitirtt dpoAryia), 
Interpolated deftnition of, 137 

Pappos: lemma 00 Apollonjus' PioHt rtisia 
O4-5: problem from same work 81: assumes 
case of VI. 3 where CKtemal angle bisected 
(Simson's vi. Prop- A) 197: theorem from 
Apollonius' Plane Loci 198; theorem that 
ratio compounded of ratios of sides is equal 
to ralio of rectangles contained by sides 
ijo: +. >7. »9. *7. 7* 81, 113, 133, 3H, 
1501 iji, i9> 

" Fturallelepipedal" (solid) numbers: two of 
the three factors differ by unity (Nicoma- 
ehus) 190 

Peletarius (Pe)etiei) : on angle ^ itntaci and 



angle of semicircle 41: 47, 56^ 84, 146, 
190 

I'entflgon : decomposition of r^nlar pentagon 
into 30 elementary triangles 98: relation to 
pentagram 99 

Pentagonal numbers 189 

"Perftct" (of a class of numbers) 193-41 
411-6: Pythagoreans applied term to tov 
194: 3 also called "perfect" 194 

Ptritirbtd pnpniitn (rtritpityiiitti dwiAffyfa) 
136. 170-7 

Pfleideier, (J. F- 1 

Philoponus 134, i8i 

Plane numbers, prtiduct of two factors 
("sides" or "length" and "breadth") 
18 7-8; in Plato either square or oblong 
187-8: similar plane numbers 193; one 
mean proportiorial between similar plane 
numb«s 371-1 

Plato: construction of r^uUr solids Irom 
triangles 97-8: op^ifm iictiim 99: 7/5 
as approximation to ^1, 119: on s<]uare 
and oolong numbers 188, 193 : on iw&fiMit 
(sqaare roots or surdi) 1881 190: theorem 
that between square numbers one mean 
suffices, between cube numbers two means 
necessary 194, 364 

Playfair, John 1 

Plutarch 98, 154 

Fariim (corollary) to proposition prccedei 
"Q.E.D," or "Q.i.r." 8, 64: Porism to iv, 
tj mentioned by Proclus 109: Porism to 
VI. 19, 134 

Polygonal numbers 189 

Powers, R. E- 4»6 

Prestet, Jean 416 

Prime (number) I definitions of, 184-5: Aris- 
totle on two senses of "prime" i8j; t 
admitted as prime by Eucl- and Aristotle, 
but excluded by Nicomachus, Theon of 
Smyrna and laroblichus, who make prime 
a subdivision of add 184-3 = "prime and 
incomposite {da^vdtroi) " 184 : different 
names for prime, " odd-times odd" (Theon), 
•'linear" (Theon), "rectilinear'' (Thy- 
maridas), "euthymetric" (lamblichns) 185: 
prime absolutely or in themselves as dis- 
tinct from prime to one another (Theon) 
18 J : definitioosof "prime to one another 
18S-6 

Proclus: on absence of formal divisions of 
proposition In certa^ cases, c-g. IV. 10, 
too: on use of "quindecagon for as* 
tronomy ill : +,39, 40, 193, 147, 169 

Proportion : complete theory applicable to 
incommensurables as well as commen- 
Eurables is due to Eudoxus iii: old 
(Pythagorean) theory practically repre- 
sented by arithmetical theory of Eucl. vii. 
113. in giving older theory as well Euclid 
simply followed tradition 113: Aristotle 
on general proof (new in his time) of 
theorem {aU^rmmda) in proportion j 13 : 
X- 5 as connecting two tnetmes 113: De 
Morgan on extension of meaning ^ nUi* 



ENGLISH INDEX 



4ii 



to cover i]Qcomm«ii$urAbLe& i j 8 : pawei of 
txprasing incom mensurable ratio is power 
of ipproiimsitioii without limit up: in- 
terpolAted definitions of proportion as 
"wmeoess" or "similarity of ratics" (19 : 
definition in V. Def. 5 substituted for that 
of VII* Def. 3o because latter found Inade- 
qttace, not vke versa i-iW De Morgan's 
defence of V. Efef. 5 as necessary and 
sulGdent 11 1-^4 x v, Def» 5 corresponds to 
Weieratiass' conception of number in 
general and to Dedekind^s theory of ir- 
ratiomds 114-fi : alternatives for v. Def. 5 
^ a geometer- friend of Saccberi, fay 
Faifofer^ Ingram!^ Veronese^ Enriques and 
Amaldi 116: proportionals of Vll. Def. 30 
{numbers) a particular case of those of v. 
Def- 5 (Sitnson^s Props. C» D and notes) 
116-9 ■ proportion in three terms (Aristotle 
makes it four) the "least'' IJi : "con- 
tinuous " proportion (irivex^ or gvvyiii^vyt 
draXayltmn Euclid J{^ iiii\ir^\ 131, i^j: 
three *' proportions ^' 391^ but proportion 
far txteiUme or primary is continuous Or 
geometric 191-3; "discrete" or "dis- 
joined" (it^^ii^ri), iit^iv^iJifii) iji, igj; 
*'ordned" proportion {-rtTay^rn^ inter- 
polated definition of, 1 37 : " perturbed " 
proportitm (rtra^^^nj) 156^ 176-7 : ex- 
tettsive use of proportions in Greek 
geometry 1 87 : proporlions enable any 
quadratic equation with real roots to he 
solved JS7: supposed use of propositions of 
Book V. io arithmetical Books 314, 310 

Fsellus ^ 

Ptolemy, Claudius: lemma about quadri- 
lateral in circle (Simson's vi. Prop. D) 
145-7 - ^I't i'7t i'9 

Pyiamidal numbers 190: pyramids truncated, 
twice-truncated etc. 191 

Pythagoras : reputed discoverer of construc- 
tion of five regular solids 97 : introduced 
" the most per^t proportion in four terms 
and specially called ^harmonic'" into 
Greece 111: construction of figure equal 
to one and similar to another rectilineal 
^ure 1J4 

Pythagoreans : construction of dodecahedron 
in sphere 97 : construction of isosceles 
tiiaogle of IV. 10 and of r^Iar pentagon 
due to, 97-S : possible method of discovery 
of latter 97-9 : theorem about only three 
regular polygons fillin|r space round a 
point ^ : distinguished three sort* of 
in^artJ, arithmetic, geometric, harmonic 
lit' had theory of proportion applicable 
to conimensurables only ill ; 7/$ as ap- 
proximation to ^3, itq: definitions of 
0011179: of even and odd 181: called lo 
" perfect" 194 

Quadratic eqtuUoDi: tolution by meini of 
proportions j8j, i6}-Ji »66-7 : iapa/iit 
or condition of possibility of solving 
equation oF Ettd. VI. iS^ijj; one solution 



only given, for obvious reasons 160, 4641 
167 : but method gives both roots if real 
158 : exact correspondence of geometrical 
to algebraical solution 163-4, i6^7 

Quadrilateral ; inscribing in circle of quadri- 
lateral equiangular to another 91-1: con- 
dition for inscribing circle in, 93, 95 : 
quadrilateral in cir^e, Ptolemy's lemma 
on (Simsmi's vi. Prop. D), 1)5-7 = quadri- 
lateral not a ' ' polygon " 139 

" Quindecagon " (fifteen- angled figure): use- 
ful for astronomy 1 1 1 

Radius: no Greek word for, 1 

Ramus, P. iji 

Ratio : definition of, 1 16-9, no sufficient 
ground for r^ardtr^ it as spurious ir7, 
Barrow's defence of it 117 : method of 
transition from arithmetical to more general 
sense covering ineommensurables ti8: 
means of ix^nssing ratio of ineommen- 
surables is by approximation to any d^re« 
of accuracy 119: def. of grater ratio only 
<?it^ criterion (there are others) 130'; tests 
for greater equal and less ratios mutually 
exclusive ijo-i : test for greater ratio 
easier to apply than that for equal ratio 
1 1^30 : a^uments about greater and less 
ratios unsafe unless they go back to original 
definitions (Simson on v- 10) 1^6-7 : rvm- 
peund T^iia 131-3, 189-90, 134: operation 
of compounding ratios 134: "ratio com- 
pounded of their sides" (careless exptes- 
sion) 14S: duplicate, trifluai€ etc. ratio 
as distinct from thuilt, tripU etc. 133: 
aliemate ratio, alUnuauU 134 : invtrs4 
ratio, inverseiy 134 ; cempojition of ratio, 
(imferundt, different from c^mpaatdiHg 
ratios i 34- j : stparaiioH of ratio, stparaitdf 
(commonly dividend^ '35: effnverJMM of 
ratio, c&HvertittdQ 1 35 : ratio ex aefuaJi 
136, ex atquaii in perturbed proporiioH 
136 ! division of ratios used in Data as 
general method alleroative to compounding 
14Q-JO ; names for particuJar arithmetical 
ratios 191 

Rtfipr0cat or redprocedly reload figures : 
definition spurious 1S9 

Riductio ad aistirdam, the only possible 
method of pioving iit. i, 8 

"Rule of three": vi. 11 equivajeat to, 11 j 

Saccberi, Getolamo (16, 130 : proof of ex- 
istence of fourth proportional by vi. i, 1, 
11, 170 
Savile, H. 190 

Sealent, a class of solid numbers 190 
Scholia: iv. No. % ascribes Book iv. to 
Pythagoreans 97 : v. No. i attributes 
Book V. to Eudoxus iti 
Scholiast to Cltuds of Aristophanes 99 
Stcth eanena attributed to Euclid 195 
Stitor (of circle) : explanation of name : two 
kinds (i) iritb vertex at centre, (1) with 
vertex at drramference j 



^ 



ENGLISH INDEX 



Secter-liii {figure} 5 : bisection of sucli a 
figure by straight line s 

S«ESii>fr, P. 416 

St^ment of circle : angle ff/^ 4 : simitar u^- 
oients ; 

Semidrcle : angle cf, 4^ 39-4 1 ('^ Angle) ** 
(tngle in semicircie a right ^ngle, pre- 
Euclidesn proof 63 

Separation of ratio, Stttipeffit XAtou, and 
leparando {AieAArrt} 1 35 : separandff and 
(omptntnds used relatively to one Hnather, 
not to oiigintil latio ifiS, 17a 

ScTTUS, C* 416 

*SlM^f of plane and soHd numbers 187-8 

Similar plane and solid numbers 993 : one 
mean between two similar plane numbers 
37i-l» two means between two similar 
solid numbers 394, 373-5 

Similar rectilineal fibres: def^ of^ given in 
Aristotle 1 88 : defTgives at once too little 
and loo much iSS: similar figures on 
straight lines which are proportional are 
themselve:s proportional and conversely 
(VI. n), alternatives for proposition 141-7 

Similar sqrnvent^ of circles 5 

Simon, Max J14, i^ 

Simpson, Thomas iii 

Simjon, R.: Props. C, D (Book v.) connect- 
ing proportioivals of vii. Def 10 as par- 
ticular case with those of V- Def^ A> 116-9: 
Anioms to Book v. tjj: Prop. B (inver- 
uon) 144: Prop. E \€ont^srisndt>) 175: 
shortens V, 8 by compressing tv^o cases 
into on« 1 5^-3 : important note showing 
daw in V. lo and giving alternative 156-7: 
Book vt. Prop. A extenJing VI. 3 to case 
where external angle bisected 197 ; Props. 
B, C, D 911-7 '• remarks on VI. i7-9i 
»S8-9 '■ '. 3. 8, ", 13 1 33t 34. ST. +J. 49. 
53. 7«. 73. 79. 9=. "7, 13", 13'. 140. 
"43-4. MS. 146. 148. IS4. 'fii. >fil. '*3. 
105, I7»^, 177, 179. rSo, (Bl, 183, 184, 
185, (86, 1S9, r93, 19s, 109, III, 111, 
i3»-t. 138. »S». ?69» *70, 171-3 

Site^ proper translation of rf^Xur^r^r in V. 
Def. J, 116-7, 189-90 

Smith and Bryant, altemarive proofs 01 v. 16, 
■7, 18 by means of vi. i, where magnitudes 
are straight lines or rectilineal areas i6j-6, 

"*9. 173-4 , , 

Solid numbers, three varieties accordiivg to 

relative length of sides loo-i 
Spktrited number, a particular kind of cube 

number 191 
Square number, product of equal numbers 

1S9, 191 : one mean between square 

numbera 194, 365-4 
Stobaeus 180 



Siibduplicale of iiny ratio found by vt. 13, 

116 
Swinden, J. H. van r8S 
Sylvester, J. 416 

Tacquet, A. ill, ij8 

Tannery, P. jm, (13 

Tartaglia, Niccolb 1, 47 

Taylor, H. M. t6, u, 19, 56, 75, 101, 117, 

i44i »47. »7' 

Tetrahedron 98 

Thales 111, 180 

Theodosius 37 

Theon of Alexandria: interpolation in v. 13 
and Porism 144 ; interpolated Porism to 
VI. 10, 139: additions to vi. 33 (about 
ttcton) 174-6: 4J, 109, 117, 119, 149, 151, 
161, 186, 190, 134, 135, 140, 141, 156, 161, 
3tt, jii, 411 

Theon of Smyrna: 111, 119, 179, iSo, 181, 
1S4, 185, 1S6, 188, 189, 19a, 191, 191, 

193, '94, 41s 

Thrasyllus 191 

Thymaridas 179, 1S5 

Timaats of Plato 97-8, 194-5, 3^3 

Todhuntcr, I., 3, 7, 11, 49. 51, 51. 67, 73, 90, 
99, i7». 19s. lo'. "4, 108, 139, 171, 171, 
300 

Ttapeiium: name applied (o truncated 
pyramidal numbers (Theon of Smyrna) 19 1 

Triangle : Heron's proof of expression (or 
area in terms of sides,s'i (i - o) (j - *) (j - f ) 
87-8 : right-angled triangle which is half 
of equilateral triangle used lor construction 
of tetrahedron, octahedron and icosahedron 
(TtmofHi of Plato) 98 

Triangular numbers ^89 

Triplitate, distinct from iriplt, ratio i^i 

at-TQsl, sa Naslraddin 

Unit: definitions of, by ThymatidaSi "some 
Pythagoreans," Chrysippus, Aristotle and 
others 179: Euclid's definition was that 
of the " more recent " writers 179 : tiiv^-f 
connected etymological !y by Theon of 
Smyrna and Nicomachus with tihvoi (soli- 
tary) or fuir-li (rest) 179 

Veronese, G. 30, 116 

Vieta : on angle if eontatt 41 

Walker 104, loS, 159 

Wallis.John: on af>f/e ^ MJiAwf (" degree 

of curvature *') 41 
Weierstrass 114 
Woepcke 5 ^ , 

Zenodorus 176 ,, .^,